text: 4 The initial part of the ﬁeld (0 < t < 2.5) is not shown in these ﬁgures as the deviations from the plane-wave so- lutions are very small and thus ﬁeld amplitude is very close to 1 for all x values. In contrast to the cubic NLSE case, the ﬁlaments of the KEE have a preferential di- rection of propagation with nonzero velocity, similar to other extension of the NLSE i.e. the Sasa-Satsumo equa- tion [6]. This is due to the fact that the group velocity of the waves is not same as the phase velocity [6]. V. STATISTICS OF BIG WAVES | | The probability distribution of amplitudes ( ψ ) in the chaotic ﬁeld provides important information about the wave ﬁeld and about the rogue waves in particu- lar. Therefore we obtain the probability density func- tions (PDFs) for various scenarios. We numerically solve the KEE and simulate the chaotic ﬁeld with in a spatial − domain of [ 1000, 1000]. We discarded the initial modu- lation instability stages in our runs and used long spatial FIG. 3: (Color online) A typical example of a chaotic wave and temporal intervals to get statistical convergence. We | | ﬁeld created by the KEE for β = 0.67 starting with modu- have divided the range of amplitudes, ψ , into 200 bins lation instability. The ﬁlaments are propagating to left with ≈ − in order to obtain relatively smooth curves and counted average group velocity v 6.5. gr the number of maxima in each bin. Then by normaliza- tion we have obtained the PDFs. For each of the PDF plots, the data we have analyzed includes approximately one million maxima which allows us to obtain relatively smooth PDFs. FIG. 4: (Color online) A typical example of a chaotic wave − ﬁeld created by the KEE for β = 0.67 starting with modu- lation instability. The ﬁlaments are propagating to right with ≈ average group velocity v 6.5. gr FIG. 5: (Color online) PDF of the KEE (—) vs. PDF of the cubic NLSE (- . -) for k = 0.1. the plane-wave can predict the magnitude of the veloc- ity of the ﬁlaments, it deems insuﬃcient to character- As Figs. 5-7 conﬁrms, the PDFs for various values ize propagation direction since the Raman-eﬀect on the of the initial plane-wavenumber (k) show that smaller plane-wave dispersion relation drops out in its derivation. values of k result in higher probability of high-amplitude However one can realize that KEE is invariant under the waves. Additionally, the comparison of PDFs of the KEE → − → − transformation β β and x x, therefore ﬁlaments and NLSE ﬁelds show that for the smaller wavenumbers, propagate in the opposite direction when the sign of the more waves emerge in the KEE ﬁeld than the NLSE ﬁeld β parameter changes. These properties of the KEE may in the amplitude interval of approximately [1, 2.2]. How- be used to model and predict the skewed shape of the ever the chaotic NLSE ﬁeld produces signiﬁcantly more rogue waves encountered in practice. waves in the amplitude interval of approximately [2.2, 5]. ##LTFigure## ##LTFigure## ##LTFigure##

text: The

text: initial

text: part

text: of

text: the

text: ﬁeld

text: t

text: is

text: not

text: shown

text: in

text: these

text: ﬁgures

text: as

text: the

text: deviations

text: from

text: the

text: plane-wave

text: so-

text: lutions

text: are

text: very

text: small

text: and

text: thus

text: ﬁeld

text: amplitude

text: is

text: very

text: close

text: to

text: for

text: all

text: x

text: values.

text: In

text: contrast

text: to

text: the

text: cubic

text: NLSE

text: case,

text: the

text: ﬁlaments

text: of

text: the

text: KEE

text: have

text: a

text: preferential

text: di-

text: rection

text: of

text: propagation

text: with

text: nonzero

text: velocity,

text: similar

text: to

text: other

text: extension

text: of

text: the

text: NLSE

text: i.e.

text: the

text: Sasa-Satsumo

text: equa-

text: tion

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: group

text: velocity

text: of

text: the

text: waves

text: is

text: not

text: same

text: as

text: the

text: phase

text: velocity

text: V.

text: STATISTICS

text: OF

text: BIG

text: WAVES

text: The

text: probability

text: distribution

text: of

text: amplitudes

text: ψ

text: in

text: the

text: chaotic

text: ﬁeld

text: provides

text: important

text: information

text: about

text: the

text: wave

text: ﬁeld

text: and

text: about

text: the

text: rogue

text: waves

text: in

text: particu-

text: lar.

text: Therefore

text: we

text: obtain

text: the

text: probability

text: density

text: func-

text: tions

text: (PDFs)

text: for

text: various

text: scenarios.

text: We

text: numerically

text: solve

text: the

text: KEE

text: and

text: simulate

text: the

text: chaotic

text: ﬁeld

text: with

text: in

text: a

text: spatial

text: domain

text: of

text: We

text: discarded

text: the

text: initial

text: modu-

text: lation

text: instability

text: stages

text: in

text: our

text: runs

text: and

text: used

text: long

text: spatial

text: FIG.

text: (Color

text: online)

text: A

text: typical

text: example

text: of

text: a

text: chaotic

text: wave

text: and

text: temporal

text: intervals

text: to

text: get

text: statistical

text: convergence.

text: We

text: ﬁeld

text: created

text: by

text: the

text: KEE

text: for

text: β

text: starting

text: with

text: modu-

text: have

text: divided

text: the

text: range

text: of

text: amplitudes,

text: ψ

text: into

text: bins

text: lation

text: instability.

text: The

text: ﬁlaments

text: are

text: propagating

text: to

text: left

text: with

text: in

text: order

text: to

text: obtain

text: relatively

text: smooth

text: curves

text: and

text: counted

text: average

text: group

text: velocity

text: v

text: gr

text: the

text: number

text: of

text: maxima

text: in

text: each

text: bin.

text: Then

text: by

text: normaliza-

text: tion

text: we

text: have

text: obtained

text: the

text: PDFs.

text: For

text: each

text: of

text: the

text: PDF

text: plots,

text: the

text: data

text: we

text: have

text: analyzed

text: includes

text: approximately

text: one

text: million

text: maxima

text: which

text: allows

text: us

text: to

text: obtain

text: relatively

text: smooth

text: PDFs.

text: FIG.

text: (Color

text: online)

text: A

text: typical

text: example

text: of

text: a

text: chaotic

text: wave

text: ﬁeld

text: created

text: by

text: the

text: KEE

text: for

text: β

text: starting

text: with

text: modu-

text: lation

text: instability.

text: The

text: ﬁlaments

text: are

text: propagating

text: to

text: right

text: with

text: average

text: group

text: velocity

text: v

text: gr

text: FIG.

text: (Color

text: online)

text: PDF

text: of

text: the

text: KEE

text: vs.

text: PDF

text: of

text: the

text: cubic

text: NLSE

text: for

text: k

text: the

text: plane-wave

text: can

text: predict

text: the

text: magnitude

text: of

text: the

text: veloc-

text: ity

text: of

text: the

text: ﬁlaments,

text: it

text: deems

text: insuﬃcient

text: to

text: character-

text: As

text: Figs.

text: conﬁrms,

text: the

text: PDFs

text: for

text: various

text: values

text: ize

text: propagation

text: direction

text: since

text: the

text: Raman-eﬀect

text: on

text: the

text: of

text: the

text: initial

text: plane-wavenumber

text: (k)

text: show

text: that

text: smaller

text: plane-wave

text: dispersion

text: relation

text: drops

text: out

text: in

text: its

text: derivation.

text: values

text: of

text: k

text: result

text: in

text: higher

text: probability

text: of

text: high-amplitude

text: However

text: one

text: can

text: realize

text: that

text: KEE

text: is

text: invariant

text: under

text: the

text: waves.

text: Additionally,

text: the

text: comparison

text: of

text: PDFs

text: of

text: the

text: KEE

text: transformation

text: β

text: β

text: and

text: x

text: x,

text: therefore

text: ﬁlaments

text: and

text: NLSE

text: ﬁelds

text: show

text: that

text: for

text: the

text: smaller

text: wavenumbers,

text: propagate

text: in

text: the

text: opposite

text: direction

text: when

text: the

text: sign

text: of

text: the

text: more

text: waves

text: emerge

text: in

text: the

text: KEE

text: ﬁeld

text: than

text: the

text: NLSE

text: ﬁeld

text: β

text: parameter

text: changes.

text: These

text: properties

text: of

text: the

text: KEE

text: may

text: in

text: the

text: amplitude

text: interval

text: of

text: approximately

text: How-

text: be

text: used

text: to

text: model

text: and

text: predict

text: the

text: skewed

text: shape

text: of

text: the

text: ever

text: the

text: chaotic

text: NLSE

text: ﬁeld

text: produces

text: signiﬁcantly

text: more

text: rogue

text: waves

text: encountered

text: in

text: practice.

text: waves

text: in

text: the

text: amplitude

text: interval

text: of

text: approximately

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: duced when using the LGP representation. Comparing the two dimensional stack approach to a single dimen- Listing 3: Code segment demonstrating an inline ad- sional stack when using an LGP representation, an aver- dition GP operator using a ﬂoat4 datatype age performance gain of 1.4x is achieved. When compar- 1 inline host device ing the LGP two dimensional GP approach to a single 2 ﬂoat4 operator+(ﬂoat4 a, ﬂoat4 b) { dimensional stack with an RPN representation, an aver- 3 4 return make ﬂoat4(a.x+b.x, a.y+b.y, age performance gain of 1.64x has now been achieved. 5 a.z+b.z, a.w+b.w) } 6 The results from using these extended data types are shown in Table 9. However, performance is only slightly 6 Further Improvements To The improved over the results from Table 8. Indeed, a perfor- GPU 2D Stack Model mance advantage is only observed for two of the problem instances whereby a ﬂoat3 data type is most eﬀective. Overall the average performance gain was only 1.01x over 6.1 Utilising SSE Data Types the results from Table 8. However, this is as expected as although there are extended data types, there are no un- derlying extended registers. Therefore there is no data The previous section has demonstrated that a two di- parallelism performed although the source code of the mensional stack model provides a signiﬁcant computa- interpreter is more predictable and this aids the CUDA tional performance advantage over a single dimensional compiler which has led to the small performance advan- stack for GPU based GP. Subsequently, each GP opera- tage that is observed. However, the use of extended data tion is now performed over multiple ﬁtness cases. This types will be of beneﬁt for the work presented in the next approach can now enable a further form of data paral- section. lelism to be used. Indeed, in the original work which im- 6.2 Using The Register File For The plemented a two dimensional stack for GP using a CPU (Chitty, 2012), the performance was extended further by Stack taking advantage of Streaming SIMD Extensions (SSE) Even though a considerable speedup has been achieved instructions. On a modern CPU there are a number of by the work presented in the previous sections, it is clear larger registers up to 256 bits wide. This enables multi- that the L1 cache memory is still under pressure restrict- ple 32 bit ﬂoats to be stored in the same registers. Sub- ing the eﬀectiveness of a two dimensional stack model for sequently, whenever an operation is performed on these GPU based GP. Subsequently, further relieving the pres- wider registers, the operation is performed upon several sure on the L1 cache memory would be favorable as it 32 bit ﬂoats simultaneously. This can be considered a may enable more ﬁtness cases to be considered in a single form of data parallelism. pass of the interpreter. To achieve this a diﬀerent mem- The GPU programming language CUDA also has a set ory resource must be considered which is not cached but of data types which can store more than one ﬂoating point operates as fast as L1 cache memory. In fact, the fastest value. These are known as ﬂoat2, ﬂoat3 and ﬂoat4 which memory resource available on a GPU is the set of registers can hold 2, 3 and 4 32 bit ﬂoats respectively. However, that are also located on-chip. Indeed, each SMX has 64k unlike a modern CPU, there are not any extra wide regis- of addressable registers available to the threads execut- ters on GPUs to enable simultaneous parallel operations ing under it which is a considerable memory storage area to be performed on these data types. Operations must but is often overlooked when implementing algorithms for still be performed in a sequential manner. However, using GPUs. In fact, Chitty (2014) demonstrated that a con- these registers can eﬀectively simplify a CUDA program siderable improvement in the speed of single dimensional and enable the source code to be more predictable to the stack GP could be achieved when utilising the GPU reg- CUDA compiler which could provide a performance ad- isters as part of the stack. vantage as instructions can then be better preloaded. A key drawback in using registers to hold the stack A ﬂoat2 data type has two 32 bit ﬂoat members x and is that they are not indexable. Instead, a conditional y. Float3 has an additional ﬂoat member z and a ﬂoat4 switch statement is required to ensure that the correct data type has a further member w. An addition opera- register is accessed when performing a stack operation. tion on two ﬂoat2 data types involves adding the two x This will inevitably add an overhead to the operation members together and then the two y members. A sim- of GP. However, since GP can be considered as much ple inline operator function can be used for GP functions memory bound as compute bound due to the high level such as addition an example of which is shown in Listing of stack operations and the limited memory resources of 3. Additionally, when accessing the stack, as both the GPUs, the beneﬁt of improved memory operations should registers and the local memory stack are deﬁned as ﬂoat4 outweigh this extra computational cost. An additional datatypes, a simple assignment statement works in the issue is that by increasing the number of registers used same manner as with a normal ﬂoat datatype. by a single thread of execution, the maximum number 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: duced

label: paragraph

text: when

text: using

text: the

text: LGP

text: representation.

text: Comparing

text: the

text: two

text: dimensional

text: stack

text: approach

text: to

text: a

text: single

text: dimen-

text: Listing

text: Code

text: segment

text: demonstrating

text: an

text: inline

text: ad-

text: sional

text: stack

text: when

text: using

text: an

text: LGP

text: representation,

text: an

text: aver-

text: dition

text: GP

text: operator

text: using

text: a

text: datatype

text: age

text: performance

text: gain

text: of

text: is

text: achieved.

text: When

text: compar-

text: inline

text: host

text: device

text: ing

text: the

text: LGP

text: two

text: dimensional

text: GP

text: approach

text: to

text: a

text: single

text: a,

text: b)

text: dimensional

text: stack

text: with

text: an

text: RPN

text: representation,

text: an

text: aver-

text: return

text: make

text: a.y+b.y,

text: age

text: performance

text: gain

text: of

text: has

text: now

text: been

text: achieved.

text: a.z+b.z,

text: a.w+b.w)

text: The

text: results

text: from

text: using

text: these

text: extended

text: data

text: types

text: are

text: shown

text: in

text: Table

text: However,

text: performance

text: is

text: only

text: slightly

text: Further

text: Improvements

text: To

text: The

text: improved

text: over

text: the

text: results

text: from

text: Table

text: Indeed,

text: a

text: perfor-

text: GPU

text: Stack

text: Model

text: mance

text: advantage

text: is

text: only

text: observed

text: for

text: two

text: of

text: the

text: problem

text: instances

text: whereby

text: a

text: data

text: type

text: is

text: most

text: eﬀective.

text: Overall

text: the

text: average

text: performance

text: gain

text: was

text: only

text: over

text: Utilising

text: SSE

text: Data

text: Types

text: the

text: results

text: from

text: Table

text: However,

text: this

text: is

text: as

text: expected

text: as

text: although

text: there

text: are

text: extended

text: data

text: types,

text: there

text: are

text: no

text: un-

text: derlying

text: extended

text: registers.

text: Therefore

text: there

text: is

text: no

text: data

text: The

text: previous

text: section

text: has

text: demonstrated

text: that

text: a

text: two

text: di-

text: parallelism

text: performed

text: although

text: the

text: source

text: code

text: of

text: the

text: mensional

text: stack

text: model

text: provides

text: a

text: signiﬁcant

text: computa-

text: interpreter

text: is

text: more

text: predictable

text: and

text: this

text: aids

text: the

text: CUDA

text: tional

text: performance

text: advantage

text: over

text: a

text: single

text: dimensional

text: compiler

text: which

text: has

text: led

text: to

text: the

text: small

text: performance

text: advan-

text: stack

text: for

text: GPU

text: based

text: GP.

text: Subsequently,

text: each

text: GP

text: opera-

text: tage

text: that

text: is

text: observed.

text: However,

text: the

text: use

text: of

text: extended

text: data

text: tion

text: is

text: now

text: performed

text: over

text: multiple

text: ﬁtness

text: cases.

text: This

text: types

text: will

text: be

text: of

text: beneﬁt

text: for

text: the

text: work

text: presented

text: in

text: the

text: next

text: approach

text: can

text: now

text: enable

text: a

text: further

text: form

text: of

text: data

text: paral-

text: section.

text: lelism

text: to

text: be

text: used.

text: Indeed,

text: in

text: the

text: original

text: work

text: which

text: im-

text: Using

text: The

text: Register

text: File

text: For

text: The

text: plemented

text: a

text: two

text: dimensional

text: stack

text: for

text: GP

text: using

text: a

text: CPU

text: (Chitty,

text: the

text: performance

text: was

text: extended

text: further

text: by

text: Stack

text: taking

text: advantage

text: of

text: Streaming

text: SIMD

text: Extensions

text: (SSE)

text: Even

text: though

text: a

text: considerable

text: speedup

text: has

text: been

text: achieved

text: instructions.

text: On

text: a

text: modern

text: CPU

text: there

text: are

text: a

text: number

text: of

text: by

text: the

text: work

text: presented

text: in

text: the

text: previous

text: sections,

text: it

text: is

text: clear

text: larger

text: registers

text: up

text: to

text: bits

text: wide.

text: This

text: enables

text: multi-

text: that

text: the

text: cache

text: memory

text: is

text: still

text: under

text: pressure

text: restrict-

text: ple

text: bit

text: ﬂoats

text: to

text: be

text: stored

text: in

text: the

text: same

text: registers.

text: Sub-

text: ing

text: the

text: eﬀectiveness

text: of

text: a

text: two

text: dimensional

text: stack

text: model

text: for

text: sequently,

text: whenever

text: an

text: operation

text: is

text: performed

text: on

text: these

text: GPU

text: based

text: GP.

text: Subsequently,

text: further

text: relieving

text: the

text: pres-

text: wider

text: registers,

text: the

text: operation

text: is

text: performed

text: upon

text: several

text: sure

text: on

text: the

text: cache

text: memory

text: would

text: be

text: favorable

text: as

text: it

text: bit

text: ﬂoats

text: simultaneously.

text: This

text: can

text: be

text: considered

text: a

text: may

text: enable

text: more

text: ﬁtness

text: cases

text: to

text: be

text: considered

text: in

text: a

text: single

text: form

text: of

text: data

text: parallelism.

text: pass

text: of

text: the

text: interpreter.

text: To

text: achieve

text: this

text: a

text: diﬀerent

text: mem-

text: The

text: GPU

text: programming

text: language

text: CUDA

text: also

text: has

text: a

text: set

text: ory

text: resource

text: must

text: be

text: considered

text: which

text: is

text: not

text: cached

text: but

text: of

text: data

text: types

text: which

text: can

text: store

text: more

text: than

text: one

text: ﬂoating

text: point

text: operates

text: as

text: fast

text: as

text: cache

text: memory.

text: In

text: fact,

text: the

text: fastest

text: value.

text: These

text: are

text: known

text: as

text: and

text: which

text: memory

text: resource

text: available

text: on

text: a

text: GPU

text: is

text: the

text: set

text: of

text: registers

text: can

text: hold

text: and

text: bit

text: ﬂoats

text: respectively.

text: However,

text: that

text: are

text: also

text: located

text: on-chip.

text: Indeed,

text: each

text: SMX

text: has

text: unlike

text: a

text: modern

text: CPU,

text: there

text: are

text: not

text: any

text: extra

text: wide

text: regis-

text: of

text: addressable

text: registers

text: available

text: to

text: the

text: threads

text: execut-

text: ters

text: on

text: GPUs

text: to

text: enable

text: simultaneous

text: parallel

text: operations

text: ing

text: under

text: it

text: which

text: is

text: a

text: considerable

text: memory

text: storage

text: area

text: to

text: be

text: performed

text: on

text: these

text: data

text: types.

text: Operations

text: must

text: but

text: is

text: often

text: overlooked

text: when

text: implementing

text: algorithms

text: for

text: still

text: be

text: performed

text: in

text: a

text: sequential

text: manner.

text: However,

text: using

text: GPUs.

text: In

text: fact,

text: Chitty

text: demonstrated

text: that

text: a

text: con-

text: these

text: registers

text: can

text: eﬀectively

text: simplify

text: a

text: CUDA

text: program

text: siderable

text: improvement

text: in

text: the

text: speed

text: of

text: single

text: dimensional

text: and

text: enable

text: the

text: source

text: code

text: to

text: be

text: more

text: predictable

text: to

text: the

text: stack

text: GP

text: could

text: be

text: achieved

text: when

text: utilising

text: the

text: GPU

text: reg-

text: CUDA

text: compiler

text: which

text: could

text: provide

text: a

text: performance

text: ad-

text: isters

text: as

text: part

text: of

text: the

text: stack.

text: vantage

text: as

text: instructions

text: can

text: then

text: be

text: better

text: preloaded.

text: A

text: key

text: drawback

text: in

text: using

text: registers

text: to

text: hold

text: the

text: stack

text: A

text: data

text: type

text: has

text: two

text: bit

text: ﬂoat

text: members

text: x

text: and

text: is

text: that

text: they

text: are

text: not

text: indexable.

text: Instead,

text: a

text: conditional

text: y.

text: has

text: an

text: additional

text: ﬂoat

text: member

text: z

text: and

text: a

text: switch

text: statement

text: is

text: required

text: to

text: ensure

text: that

text: the

text: correct

text: data

text: type

text: has

text: a

text: further

text: member

text: w.

text: An

text: addition

text: opera-

text: register

text: is

text: accessed

text: when

text: performing

text: a

text: stack

text: operation.

text: tion

text: on

text: two

text: data

text: types

text: involves

text: adding

text: the

text: two

text: x

text: This

text: will

text: inevitably

text: add

text: an

text: overhead

text: to

text: the

text: operation

text: members

text: together

text: and

text: then

text: the

text: two

text: y

text: members.

text: A

text: sim-

text: of

text: GP.

text: However,

text: since

text: GP

text: can

text: be

text: considered

text: as

text: much

text: ple

text: inline

text: operator

text: function

text: can

text: be

text: used

text: for

text: GP

text: functions

text: memory

text: bound

text: as

text: compute

text: bound

text: due

text: to

text: the

text: high

text: level

text: such

text: as

text: addition

text: an

text: example

text: of

text: which

text: is

text: shown

text: in

text: Listing

text: of

text: stack

text: operations

text: and

text: the

text: limited

text: memory

text: resources

text: of

text: Additionally,

text: when

text: accessing

text: the

text: stack,

text: as

text: both

text: the

text: GPUs,

text: the

text: beneﬁt

text: of

text: improved

text: memory

text: operations

text: should

text: registers

text: and

text: the

text: local

text: memory

text: stack

text: are

text: deﬁned

text: as

text: outweigh

text: this

text: extra

text: computational

text: cost.

text: An

text: additional

text: datatypes,

text: a

text: simple

text: assignment

text: statement

text: works

text: in

text: the

text: issue

text: is

text: that

text: by

text: increasing

text: the

text: number

text: of

text: registers

text: used

text: same

text: manner

text: as

text: with

text: a

text: normal

text: ﬂoat

text: datatype.

text: by

text: a

text: single

text: thread

text: of

text: execution,

text: the

text: maximum

text: number

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Table 4: The GPop/s (measured in billions) for each problem instance using a two-dimensional stack approach and a preference for greater L1 cache memory for a range of threads within a block and a range of second dimension sizes. The best performance for each problem instance is shown in bold. Problem Num. Size of Second Dimension Of Stack Threads 2 3 4 5 6 ± ± ± ± ± 32 13.538 0.011 16.386 0.037 18.930 0.046 18.867 0.050 19.680 0.044 ± ± ± ± ± 64 24.812 0.142 27.635 0.116 29.290 0.155 27.769 0.108 23.593 0.112 ± ± ± ± ± 96 31.192 0.144 31.598 0.208 28.971 0.144 24.686 0.071 20.506 0.046 ± ± ± ± ± 128 33.769 0.217 28.471 0.131 22.460 0.056 20.294 0.027 18.462 0.031 ± ± ± ± ± 256 34.205 0.265 29.406 0.097 22.697 0.058 21.047 0.025 18.593 0.018 ± ± ± ± ± Sextic 384 33.605 0.207 30.622 0.150 24.602 0.055 21.508 0.047 19.179 0.037 ± ± ± ± ± 512 33.398 0.202 29.744 0.113 23.490 0.051 21.684 0.031 18.545 0.021 ± ± ± ± ± 640 32.603 0.231 29.615 0.143 24.062 0.076 21.178 0.035 18.835 0.048 ± ± ± ± ± 768 29.892 0.151 29.763 0.209 27.969 0.150 25.219 0.125 20.270 0.030 ± ± ± ± ± 896 30.802 0.162 29.133 0.162 24.911 0.065 21.676 0.090 18.713 0.049 ± ± ± ± ± 1024 31.241 0.203 27.442 0.154 22.399 0.042 20.541 0.023 17.953 0.019 ± ± ± ± ± 32 13.119 0.015 17.264 0.068 20.441 0.082 20.812 0.075 22.273 0.060 ± ± ± ± ± 64 23.862 0.101 28.126 0.129 28.085 0.082 24.170 0.033 21.844 0.018 ± ± ± ± ± 96 29.719 0.108 26.686 0.043 21.416 0.027 18.873 0.019 17.439 0.014 ± ± ± ± ± 128 27.933 0.041 21.474 0.024 18.166 0.015 16.362 0.014 15.431 0.015 ± ± ± ± ± 256 29.055 0.047 21.940 0.014 18.277 0.013 16.319 0.014 15.393 0.009 ± ± ± ± ± Shuttle 384 31.168 0.100 23.712 0.027 19.293 0.016 16.920 0.026 15.621 0.010 ± ± ± ± ± 512 29.159 0.055 22.549 0.030 18.215 0.028 16.540 0.012 15.524 0.009 ± ± ± ± ± 640 30.088 0.118 22.866 0.017 18.855 0.014 16.663 0.020 15.466 0.011 ± ± ± ± ± 768 29.605 0.176 27.146 0.043 22.092 0.023 19.104 0.022 17.184 0.016 ± ± ± ± ± 896 29.813 0.106 23.394 0.022 19.002 0.015 17.031 0.017 15.696 0.013 ± ± ± ± ± 1024 27.332 0.080 21.458 0.024 17.608 0.017 15.646 0.013 14.745 0.014 ± ± ± ± ± 32 12.852 0.002 17.001 0.005 20.026 0.010 20.108 0.008 20.552 0.010 ± ± ± ± ± 64 23.939 0.008 27.235 0.026 21.661 0.014 18.920 0.006 14.849 0.007 ± ± ± ± ± 96 27.891 0.024 20.456 0.010 17.073 0.010 15.378 0.009 13.987 0.005 ± ± ± ± ± 128 21.235 0.015 17.060 0.009 15.077 0.008 13.839 0.007 13.557 0.004 ± ± ± ± ± 256 22.014 0.006 17.526 0.010 15.184 0.007 13.978 0.004 13.741 0.004 ± ± ± ± ± KDDcup 384 24.600 0.006 18.678 0.005 15.967 0.006 14.417 0.004 14.149 0.002 ± ± ± ± ± 512 22.902 0.012 18.425 0.004 15.526 0.005 14.271 0.002 14.446 0.002 ± ± ± ± ± 640 24.496 0.023 18.511 0.005 15.876 0.004 14.430 0.001 14.500 0.016 ± ± ± ± ± 768 30.536 0.013 22.203 0.005 18.373 0.006 16.198 0.003 13.845 0.005 ± ± ± ± ± 896 26.050 0.008 19.351 0.003 16.369 0.002 14.843 0.006 15.086 0.002 ± ± ± ± ± 1024 22.640 0.007 18.337 0.004 15.504 0.003 14.277 0.001 14.543 0.001 ± ± ± ± ± 32 294.742 0.302 368.967 0.716 444.253 0.456 427.078 0.882 468.940 0.659 ± ± ± ± ± 64 536.480 1.240 615.354 1.335 699.182 1.968 653.330 1.765 669.642 1.519 ± ± ± ± ± 96 697.700 1.701 730.826 1.490 744.960 0.750 677.033 0.475 653.005 0.356 ± ± ± ± ± 128 784.397 1.877 738.643 0.966 685.319 0.604 610.788 0.387 580.573 0.341 ± ± ± ± ± 256 792.898 1.547 642.027 0.644 567.665 0.435 515.719 0.464 485.477 0.331 ± ± ± ± ± 20-Mult. 384 803.756 1.958 645.628 0.625 565.830 0.406 518.172 0.390 482.823 0.351 ± ± ± ± ± 512 792.547 1.445 636.935 0.652 555.781 0.541 516.372 0.387 480.714 0.228 ± ± ± ± ± 640 792.899 2.119 647.745 1.187 562.651 0.478 495.505 0.280 478.474 0.334 ± ± ± ± ± 768 866.365 2.623 727.432 1.727 631.289 0.868 560.068 0.407 520.952 0.439 ± ± ± ± ± 896 815.287 2.381 660.618 0.809 573.290 0.489 502.213 0.476 487.595 0.191 ± ± ± ± ± 1024 771.872 1.793 620.390 0.816 547.441 0.271 485.368 0.326 451.957 0.219 This theory is reinforced by observing that as the size two sources of pressure on the L1 cache memory, the size of the second dimension of the 2D stack increases, the of the second dimension of the 2D stack model and the number of threads of execution which provide the best number of threads of execution. Consequently, a balance performance decreases. Indeed, when considering a sec- must be struck between the beneﬁts of eﬃcient interpre- ond dimension size of six, for two of the problem instances tation of candidate GP programs using a 2D stack and the best performance arises when using only 32 threads full utilisation of the SPs under each SMX. of execution resulting in most of the SPs under an SMX being idle. The amount of L1 cache memory available in each SMX is much smaller than that of a CPU. So as the number of threads of execution under each SMX multi-processor increases, the pressure on the L1 cache is increased through the use of more local memory being needed to hold the local stacks of each thread of execu- tion. However, to obtain a high level of performance for a GPU, a large number of threads are required as there are hundreds of SPs under each SMX. Thus, there are 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: The

label: caption

text: GPop/s

text: (measured

text: in

text: billions)

text: for

text: each

text: problem

text: instance

text: using

text: a

text: two-dimensional

text: stack

text: approach

text: and

text: a

text: preference

text: for

text: greater

text: cache

text: memory

text: for

text: a

text: range

text: of

text: threads

text: within

text: a

text: block

text: and

text: a

text: range

text: of

text: second

text: dimension

text: sizes.

text: The

text: best

text: performance

text: for

text: each

text: problem

text: instance

text: is

text: shown

text: in

text: bold.

text: Problem

label: table

text: Num.

text: Size

text: of

text: Second

text: Dimension

text: Of

text: Stack

text: Threads

text: Sextic

text: Shuttle

text: KDDcup

text: This

text: theory

text: is

text: reinforced

text: by

text: observing

text: that

text: as

text: the

text: size

text: two

text: sources

text: of

text: pressure

text: on

text: the

text: cache

text: memory,

text: the

text: size

text: of

text: the

text: second

text: dimension

text: of

text: the

text: stack

text: increases,

text: the

text: of

text: the

text: second

text: dimension

text: of

text: the

text: stack

text: model

text: and

text: the

text: number

text: of

text: threads

text: of

text: execution

text: which

text: provide

text: the

text: best

text: number

text: of

text: threads

text: of

text: execution.

text: Consequently,

text: a

text: balance

text: performance

text: decreases.

text: Indeed,

text: when

text: considering

text: a

text: sec-

text: must

text: be

text: struck

text: between

text: the

text: beneﬁts

text: of

text: eﬃcient

text: interpre-

text: ond

text: dimension

text: size

text: of

text: six,

text: for

text: two

text: of

text: the

text: problem

text: instances

text: tation

text: of

text: candidate

text: GP

text: programs

text: using

text: a

text: stack

text: and

text: the

text: best

text: performance

text: arises

text: when

text: using

text: only

text: threads

text: full

text: utilisation

text: of

text: the

text: SPs

text: under

text: each

text: SMX.

text: of

text: execution

text: resulting

text: in

text: most

text: of

text: the

text: SPs

text: under

text: an

text: SMX

text: being

text: idle.

text: The

text: amount

text: of

text: cache

text: memory

text: available

text: in

text: each

text: SMX

text: is

text: much

text: smaller

text: than

text: that

text: of

text: a

text: CPU.

text: So

text: as

text: the

text: number

text: of

text: threads

text: of

text: execution

text: under

text: each

text: SMX

text: multi-processor

text: increases,

text: the

text: pressure

text: on

text: the

text: cache

text: is

text: increased

text: through

text: the

text: use

text: of

text: more

text: local

text: memory

text: being

text: needed

text: to

text: hold

text: the

text: local

text: stacks

text: of

text: each

text: thread

text: of

text: execu-

text: tion.

text: However,

text: to

text: obtain

text: a

text: high

text: level

text: of

text: performance

text: for

text: a

text: GPU,

text: a

text: large

text: number

text: of

text: threads

text: are

text: required

text: as

text: there

text: are

text: hundreds

text: of

text: SPs

text: under

text: each

text: SMX.

text: Thus,

text: there

text: are

text: ##LTLine##

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text: 12 FRICAIN, HARTMANN, AND ROSS (ii) there exists a δ > 0 and an η > 0 such that | ∩ − | (cid:62) ∀ ∈ R (5.2) E [x η, x + η] δ, x . Notice how condition (ii) is a relative density condition we have met before when studying the Bergman and Fock spaces. R Lin [32] generalized the above result for measures µ on . We say that a positive locally ﬁnite R measure µ on is h-equivalent to Lebesgue measure if there exists a K > 0 such that − (cid:16) ∀ ∈ R | | µ(x h, x + h) h x , x > K. Theorem 5.3. Suppose µ is a locally ﬁnite Borel measure on R. (cid:90) (cid:90) (i) There exists a constant γ > 0 such that if µ is h-equivalent to Lebesgue measure for some h < γ then | | (cid:16) | | ∀ ∈ F 2dt F 2dµ F P W. (cid:90)R (cid:90)R (ii) If | | (cid:16) | | ∀ ∈ F 2dt F 2dµ F P W, R R then µ is h-equivalent to Lebesgue measure for some h > 0. 6. MODEL SPACES D A bounded analytic function Θ on is called an inner function if the radial limits of Θ (which T exist almost everywhere on [17]) are unimodular almost everywhere. Examples of inner func- ⊂ D tions include the Blaschke products B with (Blaschke) zeros Λ and singular inner functions Λ T with associated (positive) singular measure ν on . In fact, every inner function is a product of these two basic types [17]. (cid:26) (cid:90) (cid:27) Associated to each inner function Θ is a model space K ⊥ ∈ ∀ ∈ := (ΘH2) = f H2 : fΘgdm = 0 g H2 . Θ T Model spaces are the generic (closed) invariant subspaces of H2 for the backward shift operator − ∗ f(z) f(0) (S f)(z) = . z Moreover, the compression of the shift operator (Sf)(z) = zf(z) to a model space is the so-called “model operator” for certain types of Hilbert space contractions. ##LTLine## ##LTLine##

text: FRICAIN,

text: HARTMANN,

text: AND

text: ROSS

text: (ii)

label: list

text: there

text: exists

text: a

text: δ

text: and

text: an

text: η

text: such

text: that

label: equation

text: R

text: E

text: [x

text: η,

text: x

text: η]

text: δ,

text: x

text: Notice

text: how

text: condition

text: (ii)

text: is

text: a

text: relative

text: density

text: condition

text: we

text: have

text: met

text: before

text: when

text: studying

text: the

text: Bergman

text: and

text: Fock

text: spaces.

text: R

text: Lin

text: generalized

text: the

text: above

text: result

text: for

text: measures

text: µ

text: on

text: We

text: say

text: that

text: a

text: positive

text: locally

text: ﬁnite

text: R

text: measure

text: µ

text: on

text: is

text: h-equivalent

text: to

text: Lebesgue

text: measure

text: if

text: there

text: exists

text: a

text: K

text: such

text: that

text: R

text: µ(x

text: h,

text: x

text: h)

text: h

text: x

text: x

text: K.

text: Theorem

text: Suppose

text: µ

text: is

text: a

text: locally

text: ﬁnite

text: Borel

text: measure

text: on

text: R.

text: (i)

text: There

text: exists

text: a

text: constant

text: γ

text: such

text: that

text: if

text: µ

text: is

text: h-equivalent

text: to

text: Lebesgue

text: measure

text: for

text: some

text: h

text: γ

text: then

text: F

text: F

text: F

text: P

text: W.

text: (ii)

text: If

text: F

text: F

text: F

text: P

text: W,

text: R

text: R

text: then

text: µ

text: is

text: h-equivalent

text: to

text: Lebesgue

text: measure

text: for

text: some

text: h

text: MODEL

text: SPACES

text: D

text: A

text: bounded

text: analytic

text: function

text: Θ

text: on

text: is

text: called

text: an

text: inner

text: function

text: if

text: the

text: radial

text: limits

text: of

text: Θ

text: (which

text: T

text: exist

text: almost

text: everywhere

text: on

text: are

text: unimodular

text: almost

text: everywhere.

text: Examples

text: of

text: inner

text: func-

text: D

text: tions

text: include

text: the

text: Blaschke

text: products

text: B

text: with

text: (Blaschke)

text: zeros

text: Λ

text: and

text: singular

text: inner

text: functions

text: Λ

text: T

text: with

text: associated

text: (positive)

text: singular

text: measure

text: ν

text: on

text: In

text: fact,

text: every

text: inner

text: function

text: is

text: a

text: product

text: of

text: these

text: two

text: basic

text: types

text: Associated

text: to

text: each

text: inner

text: function

text: Θ

text: is

text: a

text: model

text: space

text: K

text: f

text: fΘgdm

text: g

text: Θ

text: T

text: Model

text: spaces

text: are

text: the

text: generic

text: (closed)

text: invariant

text: subspaces

text: of

text: for

text: the

text: backward

text: shift

text: operator

text: f(z)

text: (S

text: f)(z)

text: z

text: Moreover,

text: the

text: compression

text: of

text: the

text: shift

text: operator

text: (Sf)(z)

text: zf(z)

text: to

text: a

text: model

text: space

text: is

text: the

text: so-called

text: “model

text: operator”

text: for

text: certain

text: types

text: of

text: Hilbert

text: space

text: contractions.

text: ##LTLine##

text: ##LTLine##


text: A SURVEY ON REVERSE CARLESON MEASURES 13 (cid:18) (cid:19) It turns out that the Paley-Wiener space P W can be viewed as a certain type of model space. We follow [48]. Let z + 1 Ψ(z) := exp 2π − (cid:90) z 1 be the atomic inner function with point mass at z = 1 and with weight 2π, √1 − (F f)(x) := e ixtf(t)dt, 2π R (cid:0) (cid:1) R the Fourier transform on L2( ), and − → R √1 1 x i J : L2(m) L2( ), (Jg)(x) = f . π x + i x + i R It is well known that F is a unitary operator on L2( ) and a change of variables will show that R J is a unitary map from L2(m) onto L2( ). It is also known [48, p. 33] that K (F J) = L2[0, 2π]. Ψ If → − T : L2[0, 2π] L2[ π, π], (T h)(x) = h(x + π) is the translation operator then K − (T F J) = L2[ π, π] Ψ and K (F T F J) = P W. Ψ Thus the Paley-Wiener space is an isometric copy of a certain model space in a prescribed way. An important set associated with an inner function is its boundary spectrum (cid:26) (cid:27) ∈ T | | (6.1) σ(Θ) := ξ : lim Θ (z) = 0 . → z ξ Using the factorization of Θ into a Blaschke product and a singular inner function, one can show (cid:54) T T \ that when σ(Θ) = , there is a two-dimensional open neighborhood Ω containing σ(Θ) such that Θ has an analytic continuation to Ω. Functions in model spaces can have more regularity than generic functions in H2. Indeed, a K result of Moeller [38] says every function in follows the behavior of its corresponding inner Θ T \ functions and has an analytic continuation to a two dimensional open neighborhood of σ(Θ). ∈ T \ In fact, one can say a little bit more. Indeed, fo(cid:112)r every ξ σ(Θ) the evaluation functional K E f = f(ξ) is continuous on with ξ Θ (cid:107) (cid:107) | (cid:48) | E = Θ (ξ) . ξ Thus ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

text: It

text: turns

text: out

text: that

text: the

text: Paley-Wiener

text: space

text: P

text: W

text: can

text: be

text: viewed

text: as

text: a

text: certain

text: type

text: of

text: model

text: space.

text: We

text: follow

text: Let

text: z

text: Ψ(z)

text: exp

text: 2π

text: z

text: be

text: the

text: atomic

text: inner

text: function

text: with

text: point

text: mass

text: at

text: z

text: and

text: with

text: weight

text: √1

text: (F

text: f)(x)

text: e

text: ixtf(t)dt,

text: 2π

text: R

text: R

text: the

text: Fourier

text: transform

text: on

text: and

text: R

text: √1

text: x

text: i

text: J

text: (Jg)(x)

text: f

text: π

text: x

text: i

text: x

text: i

text: R

text: It

text: is

text: well

text: known

text: that

text: F

text: is

text: a

text: unitary

text: operator

text: on

text: and

text: a

text: change

text: of

text: variables

text: will

text: show

text: that

text: R

text: J

text: is

text: a

text: unitary

text: map

text: from

text: onto

text: It

text: is

text: also

text: known

text: p.

text: that

text: K

text: (F

text: J)

text: Ψ

text: If

text: T

text: π,

text: π],

text: (T

text: h)(x)

text: h(x

text: π)

text: is

text: the

text: translation

text: operator

text: then

text: K

text: (T

text: F

text: J)

text: π,

text: π]

text: Ψ

text: and

text: K

text: (F

text: T

text: F

text: J)

text: P

text: W.

text: Ψ

text: Thus

text: the

text: Paley-Wiener

text: space

text: is

text: an

text: isometric

text: copy

text: of

text: a

text: certain

text: model

text: space

text: in

text: a

text: prescribed

text: way.

text: An

text: important

text: set

text: associated

text: with

text: an

text: inner

text: function

text: is

text: its

text: boundary

text: spectrum

text: T

text: σ(Θ)

text: ξ

text: lim

text: Θ

text: (z)

text: z

text: ξ

text: Using

text: the

text: factorization

text: of

text: Θ

text: into

text: a

text: Blaschke

text: product

text: and

text: a

text: singular

text: inner

text: function,

text: one

text: can

text: show

text: T

text: T

text: that

text: when

text: σ(Θ)

text: there

text: is

text: a

text: two-dimensional

text: open

text: neighborhood

text: Ω

text: containing

text: σ(Θ)

text: such

text: that

text: Θ

text: has

text: an

text: analytic

text: continuation

text: to

text: Ω.

text: Functions

text: in

text: model

text: spaces

text: can

text: have

text: more

text: regularity

text: than

text: generic

text: functions

text: in

text: Indeed,

text: a

text: K

text: result

text: of

text: Moeller

text: says

text: every

text: function

text: in

text: follows

text: the

text: behavior

text: of

text: its

text: corresponding

text: inner

text: Θ

text: T

text: functions

text: and

text: has

text: an

text: analytic

text: continuation

text: to

text: a

text: two

text: dimensional

text: open

text: neighborhood

text: of

text: σ(Θ).

text: T

text: In

text: fact,

text: one

text: can

text: say

text: a

text: little

text: bit

text: more.

text: Indeed,

text: every

text: ξ

text: σ(Θ)

text: the

text: evaluation

text: functional

text: K

text: E

text: f

text: f(ξ)

text: is

text: continuous

text: on

text: with

text: ξ

text: Θ

text: E

text: Θ

text: (ξ)

text: ξ

text: Thus

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A SURVEY ON REVERSE CARLESON MEASURES 15 ∈ D− K µ M ( ) and (6.5) holds, then µ is Θ-admissible, every function in has radial limits | + T (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ K Θ µ T-almost everywhere on , and f f for every f . µ m Θ K Carleson measures for were discussed in the papers of Cohn [14] and Treil and Volberg [55]. Θ Their theorem is stated in terms of { ∈ D | | } (6.6) Ω(Θ, ε) := z : Θ(z) < ε , 0 < ε < 1, the sub-level sets for Θ. Note that boundary spectrum σ(Θ) is contained in the closure of any Ω(Θ, ε), 0 < ε < 1. Theorem 6.7. Suppose µ ∈ M (D−) and deﬁne the following conditions: + (cid:46) | | ⊂ T ∩ (cid:54) ∅ (i) µ(S ) I for all arcs I for which S Ω(Θ, ε) = ; I I K (ii) µ is a Carleson measure for ; Θ (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ D (iii) µ is Θ-admissible and kΘ kΘ holds for every λ . λ µ λ m ⇒ ⇒ ∈ Then (i) = (ii) = (iii). Moreover, if for some ε (0, 1), the sub-level set Ω(Θ, ε) is ⇐⇒ ⇐⇒ connected, then (i) (ii) (iii). ∈ The condition that Ω(Θ, ε) is connected for some ε (0, 1) is often called the connected level ∈ set condition (CLS). Cohn [14] proved that if Ω((cid:18)Θ, ε) is (cid:19)connected and δ (ε, 1), then Ω(Θ, δ) is also connected. Any ﬁnite Blaschke product, the atomic inner function z + 1 Θ(z) = exp − , z 1 { − } and the inﬁnite Blaschke product whose zeros are 1 rn (cid:62) , where 0 < r < 1, satisfy this n 1 connected level set condition. The sufﬁcient condition appearing in assertion (i) of Theorem 6.7 is, in general, not necessary. More precisely, Treil and Volberg [55] proved that this condition is necessary for the embedding K ∈ of into L2(µ) if and only if Θ (CLS). Nazarov–Volberg [39] proved that the RKT (repro- Θ K ducing kernel thesis) for Carleson embeddings for is, in general, not true. In [3], Baranov Θ obtained a signiﬁcant extension of the Cohn and Volberg–Treil results, introducing a new point of view based on certain Bernstein-type inequalities. Quite recently, in answering a question posed by Sarason [52], Baranov–Besonnov–Kapustin [6] clariﬁed a nice link between Carleson K measures for and an interesting class of operators – the truncated Toeplitz operators – which Θ have received much attention in the last few years [52]. We turn to reverse Carleson measures. Since the main reverse embedding result for model spaces, or backward shift invariant subspaces, is new in the non Hilbert situation we will state this theo- ∞ rem for 1 < p < + . In this more general situation we need the following deﬁnition K ∩ p = Hp ΘHp, Θ 0 ##LTLine## ##LTLine##

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

text: D−

text: K

text: µ

text: M

text: and

text: holds,

text: then

text: µ

text: is

text: Θ-admissible,

text: every

text: function

text: in

text: has

text: radial

text: limits

text: T

text: K

text: Θ

text: µ

text: T-almost

text: everywhere

text: on

text: and

text: f

text: f

text: for

text: every

text: f

text: µ

text: m

text: Θ

text: K

text: Carleson

text: measures

text: for

text: were

text: discussed

text: in

text: the

text: papers

text: of

text: Cohn

text: and

text: Treil

text: and

text: Volberg

text: Θ

text: Their

text: theorem

text: is

text: stated

text: in

text: terms

text: of

text: D

text: Ω(Θ,

text: ε)

text: z

text: Θ(z)

text: ε

text: ε

text: the

text: sub-level

text: sets

text: for

text: Θ.

text: Note

text: that

text: boundary

text: spectrum

text: σ(Θ)

text: is

text: contained

text: in

text: the

text: closure

text: of

text: any

text: Ω(Θ,

text: ε),

text: ε

text: Theorem

text: Suppose

text: µ

text: M

text: (D−)

text: and

text: deﬁne

text: the

text: following

text: conditions:

text: T

text: (i)

text: µ(S

text: I

text: for

text: all

text: arcs

text: I

text: for

text: which

text: S

text: Ω(Θ,

text: ε)

text: I

text: I

text: K

text: (ii)

text: µ

text: is

text: a

text: Carleson

text: measure

text: for

text: Θ

text: D

text: (iii)

text: µ

text: is

text: Θ-admissible

text: and

text: kΘ

text: kΘ

text: holds

text: for

text: every

text: λ

text: λ

text: µ

text: λ

text: m

text: Then

text: (i)

text: (ii)

text: (iii).

text: Moreover,

text: if

text: for

text: some

text: ε

text: the

text: sub-level

text: set

text: Ω(Θ,

text: ε)

text: is

text: connected,

text: then

text: (i)

text: (ii)

text: (iii).

text: The

text: condition

text: that

text: Ω(Θ,

text: ε)

text: is

text: connected

text: for

text: some

text: ε

text: is

text: often

text: called

text: the

text: connected

text: level

text: set

text: condition

text: (CLS).

text: Cohn

text: proved

text: that

text: if

text: ε)

text: is

text: and

text: δ

text: (ε,

text: then

text: Ω(Θ,

text: δ)

text: is

text: also

text: connected.

text: Any

text: ﬁnite

text: Blaschke

text: product,

text: the

text: atomic

text: inner

text: function

text: z

text: Θ(z)

text: exp

text: z

text: and

text: the

text: inﬁnite

text: Blaschke

text: product

text: whose

text: zeros

text: are

text: rn

text: where

text: r

text: satisfy

text: this

text: n

text: connected

text: level

text: set

text: condition.

text: The

text: sufﬁcient

text: condition

text: appearing

text: in

text: assertion

text: (i)

text: of

text: Theorem

text: is,

text: in

text: general,

text: not

text: necessary.

text: More

text: precisely,

text: Treil

text: and

text: Volberg

text: proved

text: that

text: this

text: condition

text: is

text: necessary

text: for

text: the

text: embedding

text: K

text: of

text: into

text: if

text: and

text: only

text: if

text: Θ

text: (CLS).

text: Nazarov–Volberg

text: proved

text: that

text: the

text: RKT

text: (repro-

text: Θ

text: K

text: ducing

text: kernel

text: thesis)

text: for

text: Carleson

text: embeddings

text: for

text: is,

text: in

text: general,

text: not

text: true.

text: In

text: Baranov

text: Θ

text: obtained

text: a

text: signiﬁcant

text: extension

text: of

text: the

text: Cohn

text: and

text: Volberg–Treil

text: results,

text: introducing

text: a

text: new

text: point

text: of

text: view

text: based

text: on

text: certain

text: Bernstein-type

text: inequalities.

text: Quite

text: recently,

text: in

text: answering

text: a

text: question

text: posed

text: by

text: Sarason

text: Baranov–Besonnov–Kapustin

text: clariﬁed

text: a

text: nice

text: link

text: between

text: Carleson

text: K

text: measures

text: for

text: and

text: an

text: interesting

text: class

text: of

text: operators

text: the

text: truncated

text: Toeplitz

text: operators

text: which

text: Θ

text: have

text: received

text: much

text: attention

text: in

text: the

text: last

text: few

text: years

text: We

text: turn

text: to

text: reverse

text: Carleson

text: measures.

text: Since

text: the

text: main

text: reverse

text: embedding

text: result

text: for

text: model

text: spaces,

text: or

text: backward

text: shift

text: invariant

text: subspaces,

text: is

text: new

text: in

text: the

text: non

text: Hilbert

text: situation

text: we

text: will

text: state

text: this

text: theo-

text: rem

text: for

text: p

text: In

text: this

text: more

text: general

text: situation

text: we

text: need

text: the

text: following

text: deﬁnition

text: K

text: p

text: Hp

text: ΘHp,

text: Θ

text: ##LTLine##

text: ##LTLine##


text: A SURVEY ON REVERSE CARLESON MEASURES 17 K This is equivalent to saying that the measure dµ = χ dm is a reverse Carleson measure for . Σ Θ Here we list some observations concerning dominating sets for model spaces. We will use the following notation for sets A, B and a point x: {| − | ∈ ∈ } { } d(A, B) := inf a b : a A, b B , d(x, A) := d( x , A). Throughout the list below we will assume that Θ is inner and σ(Θ) is its boundary spectrum from (6.1). All of these results can be found in [8, Section 5]. K ∈ (i) If Σ is a dominating set for then, for every ζ σ(Θ), we have d(ζ, Σ) = 0. Θ K (ii) If Σ is a dominating set for then d(Σ, σ(Θ)) = 0. Θ ∈ (iii) Let ζ σ(Θ) and Σ dominating. Then there exists an α > 0 such that for every sequence → → λ ζ with Θ(λ ) 0, there is an integer N with n n ∩ (cid:38) (cid:62) m(Σ Iα ) m(Iα ), n N. λn λn T − | | In the above, Iα is the subarc of centered at |λ| with length α(1 λ ). λ λ T ⊂ K (iv) Every open subset Σ of such that σ(Θ) Σ and m(Σ) < 1 is a dominating set for . Θ ∈ (v) Let Θ be an inner function such that m(σ(Θ)) = 0. Then for every ε (0, 1) there is a K dominating set Σ for such that m(Σ) < ε. In particular, this is true for (CLS)-inner Θ functions. T K T (vi) If σ(Θ) = and if Σ is a dominating set for then Σ is dense in . Θ T (cid:40) T (vii) There exists a Blaschke product B with σ(B) = and an open subset Σ dominating K for . B (vi) Every model space admits a dominating set. Theorem 6.8 shows, in the special case of the Paley-Wiener space, that when (5.2) is satisﬁed for sufﬁciently small η, then E is a dominating set for P W . For reverse Carleson measures there is the following result from [8]. Theorem 6.13. Let Θ be an inner function, Σ be a dominating set for K , and µ ∈ M (D−). Θ + Suppose that µ(S ) I inf > 0, I m (I) ⊂ T ∩ (cid:54) ∅ where the above inﬁmum is taken over all arcs I such that I Σ = . Then (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∀ ∈ K ∩ D− (6.14) f f f C( ). m µ Θ Corollary 6.15. Under the hypotheses of Theorem 6.13, and if moreover the measure µ is as- K sumed to be Θ-admissible, then the inequality in (6.14) extends to all of . Θ ##LTLine## ##LTLine##

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

text: K

text: This

text: is

text: equivalent

text: to

text: saying

text: that

text: the

text: measure

text: dµ

text: χ

text: dm

text: is

text: a

text: reverse

text: Carleson

text: measure

text: for

text: Σ

text: Θ

text: Here

text: we

text: list

text: some

text: observations

text: concerning

text: dominating

text: sets

text: for

text: model

text: spaces.

text: We

text: will

text: use

text: the

text: following

text: notation

text: for

text: sets

text: A,

text: B

text: and

text: a

text: point

text: x:

text: d(A,

text: B)

text: inf

text: a

text: b

text: a

text: A,

text: b

text: B

text: d(x,

text: A)

text: d(

text: x

text: A).

text: Throughout

text: the

text: list

text: below

text: we

text: will

text: assume

text: that

text: Θ

text: is

text: inner

text: and

text: σ(Θ)

text: is

text: its

text: boundary

text: spectrum

text: from

text: All

text: of

text: these

text: results

text: can

text: be

text: found

text: in

text: Section

text: K

text: (i)

text: If

text: Σ

text: is

text: a

text: dominating

text: set

text: for

text: then,

text: for

text: every

text: ζ

text: σ(Θ),

text: we

text: have

text: d(ζ,

text: Σ)

text: Θ

text: K

text: (ii)

text: If

text: Σ

text: is

text: a

text: dominating

text: set

text: for

text: then

text: d(Σ,

text: σ(Θ))

text: Θ

text: (iii)

text: Let

text: ζ

text: σ(Θ)

text: and

text: Σ

text: dominating.

text: Then

text: there

text: exists

text: an

text: α

text: such

text: that

text: for

text: every

text: sequence

text: λ

text: ζ

text: with

text: Θ(λ

text: there

text: is

text: an

text: integer

text: N

text: with

text: n

text: n

text: m(Σ

text: Iα

text: m(Iα

text: n

text: N.

text: λn

text: λn

text: T

text: In

text: the

text: above,

text: Iα

text: is

text: the

text: subarc

text: of

text: centered

text: at

text: |λ|

text: with

text: length

text: λ

text: λ

text: λ

text: T

text: K

text: (iv)

text: Every

text: open

text: subset

text: Σ

text: of

text: such

text: that

text: σ(Θ)

text: Σ

text: and

text: m(Σ)

text: is

text: a

text: dominating

text: set

text: for

text: Θ

text: (v)

text: Let

text: Θ

text: be

text: an

text: inner

text: function

text: such

text: that

text: m(σ(Θ))

text: Then

text: for

text: every

text: ε

text: there

text: is

text: a

text: K

text: dominating

text: set

text: Σ

text: for

text: such

text: that

text: m(Σ)

text: ε.

text: In

text: particular,

text: this

text: is

text: true

text: for

text: (CLS)-inner

text: Θ

text: functions.

text: T

text: K

text: T

text: (vi)

text: If

text: σ(Θ)

text: and

text: if

text: Σ

text: is

text: a

text: dominating

text: set

text: for

text: then

text: Σ

text: is

text: dense

text: in

text: Θ

text: T

text: T

text: (vii)

text: There

text: exists

text: a

text: Blaschke

text: product

text: B

text: with

text: σ(B)

text: and

text: an

text: open

text: subset

text: Σ

text: dominating

text: K

text: for

text: B

text: (vi)

text: Every

text: model

text: space

text: admits

text: a

text: dominating

text: set.

text: Theorem

text: shows,

text: in

text: the

text: special

text: case

text: of

text: the

text: Paley-Wiener

text: space,

text: that

text: when

text: is

text: satisﬁed

text: for

text: sufﬁciently

text: small

text: η,

text: then

text: E

text: is

text: a

text: dominating

text: set

text: for

text: P

text: W

text: For

text: reverse

text: Carleson

text: measures

text: there

text: is

text: the

text: following

text: result

text: from

text: Theorem

text: Let

text: Θ

text: be

text: an

text: inner

text: function,

text: Σ

text: be

text: a

text: dominating

text: set

text: for

text: K

text: and

text: µ

text: M

text: (D−).

text: Θ

text: Suppose

text: that

text: µ(S

text: I

text: inf

text: I

text: m

text: (I)

text: T

text: where

text: the

text: above

text: inﬁmum

text: is

text: taken

text: over

text: all

text: arcs

text: I

text: such

text: that

text: I

text: Σ

text: Then

text: K

text: D−

text: f

text: f

text: f

text: C(

text: m

text: µ

text: Θ

text: Corollary

text: Under

text: the

text: hypotheses

text: of

text: Theorem

text: and

text: if

text: moreover

text: the

text: measure

text: µ

text: is

text: as-

text: K

text: sumed

text: to

text: be

text: Θ-admissible,

text: then

text: the

text: inequality

text: in

text: extends

text: to

text: all

text: of

text: Θ

text: ##LTLine##

text: ##LTLine##


text: 18 FRICAIN, HARTMANN, AND ROSS For the Hardy space, the reverse Carleson measures were characterized by the reverse reproduc- (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ D ing kernel thesis, i.e., k k for all λ . For model spaces, however, the reverse λ m λ µ reproducing kernel thesis is a spectacular failure [23]. Theorem 6.16. Let Θ be an inner function that is not a ﬁnite Blaschke product. Then there ∈ T K exists a measure µ M ( ) such that µ is a Carleson measure for , the reverse estimate on + Θ reproducing kernels kΘ, λ (cid:107) (cid:107) (cid:38) (cid:107) (cid:107) ∀ ∈ D kΘ kΘ λ , λ µ λ m K is satisﬁed, but µ is not a reverse Carleson measure for . Θ Let us see this counterexample worked out in th(cid:0)e special (cid:1)case of the Paley-Wiener space P W , K which, recall from our earlier discussion, is isometrically isomorphic to the model space with Θ z + 1 Θ(z) = exp 2π − . (cid:40) z 1 { } Consider the sequence S = x ∈Z\{ }, where n n 0 n + 1/8 if n is even x = − n n 1/8 if n is odd. By the Kadets-Ingham theorem [40, Theorem(cid:88)D4.1.2], S is a minimal sampling (or complete interpolating) sequence if we include the point 0. Since S is not sampling, the discrete measure µ := δ xn (cid:54) n=0 does not satisfy the reverse inequality (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∀ ∈ f R f f P W. L2( ) L2(µ) However, the L2(µ)-norm of the normalized reproducing kernels − − sin(π(z λ)) (cid:39) | | − | | K (z) = c sinc(π(z λ)) = c − , c2 (1 + Im λ )e 2π Imλ , λ λ λ π(z λ) λ | | is uniformly bounded from below. Indeed, if λ is such that Im λ > 1 then | − | (cid:39) | | (cid:90) sin(π(x(cid:12) λ)) eπ Im(cid:12) λ , (cid:88) (cid:12) n (cid:12) (cid:88) and hence (cid:12) (cid:12) (cid:12) (cid:12) − | | | | sin(π(x λ)) 2 (cid:39) Im λ (cid:39) K (x) 2dµ(x) = c2 n− | − | 1. C λ (cid:54) λ π(xn λ) (cid:54) xn λ 2 n=0 n=0 ∈ C | | (cid:54) Thus it is enough(cid:90)to consider points λ with Im λ (cid:12) 1. Let x be t(cid:12)he point of S closest to (cid:88) (cid:12) n0 (cid:12) λ. Then there is δ > 0, independent of λ, such that (cid:12) (cid:12) (cid:12) (cid:12) − | | | | (cid:62) sin(π(x λ)) 2 (cid:62) K (x) 2dµ(x) = K (x ) 2 n0− δ. λ λ n C n(cid:54)=0 π(xn0 λ) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FRICAIN,

text: HARTMANN,

text: AND

text: ROSS

text: For

text: the

text: Hardy

text: space,

text: the

text: reverse

text: Carleson

text: measures

text: were

text: characterized

text: by

text: the

text: reverse

text: reproduc-

text: D

text: ing

text: kernel

text: thesis,

text: i.e.,

text: k

text: k

text: for

text: all

text: λ

text: For

text: model

text: spaces,

text: however,

text: the

text: reverse

text: λ

text: m

text: λ

text: µ

text: reproducing

text: kernel

text: thesis

text: is

text: a

text: spectacular

text: failure

text: Theorem

text: Let

text: Θ

text: be

text: an

text: inner

text: function

text: that

text: is

text: not

text: a

text: ﬁnite

text: Blaschke

text: product.

text: Then

text: there

text: T

text: K

text: exists

text: a

text: measure

text: µ

text: M

text: such

text: that

text: µ

text: is

text: a

text: Carleson

text: measure

text: for

text: the

text: reverse

text: estimate

text: on

text: Θ

text: reproducing

text: kernels

text: kΘ,

text: λ

text: D

text: kΘ

text: kΘ

text: λ

text: λ

text: µ

text: λ

text: m

text: K

text: is

text: satisﬁed,

text: but

text: µ

text: is

text: not

text: a

text: reverse

text: Carleson

text: measure

text: for

text: Θ

text: Let

text: us

text: see

text: this

text: counterexample

text: worked

text: out

text: in

text: special

text: of

text: the

text: Paley-Wiener

text: space

text: P

text: W

text: K

text: which,

text: recall

text: from

text: our

text: earlier

text: discussion,

text: is

text: isometrically

text: isomorphic

text: to

text: the

text: model

text: space

text: with

text: Θ

text: z

text: Θ(z)

text: exp

text: 2π

text: z

text: Consider

text: the

text: sequence

text: S

text: x

text: ∈Z\{

text: where

text: n

text: n

text: n

text: if

text: n

text: is

text: even

text: x

text: n

text: n

text: if

text: n

text: is

text: odd.

text: By

text: the

text: Kadets-Ingham

text: theorem

text: S

text: is

text: a

text: minimal

text: sampling

text: (or

text: complete

text: interpolating)

text: sequence

text: if

text: we

text: include

text: the

text: point

text: Since

text: S

text: is

text: not

text: sampling,

text: the

text: discrete

text: measure

text: µ

text: δ

text: xn

text: does

text: not

text: satisfy

text: the

text: reverse

text: inequality

text: f

text: R

text: f

text: f

text: P

text: W.

text: However,

text: the

text: of

text: the

text: normalized

text: reproducing

text: kernels

text: sin(π(z

text: λ))

text: K

text: (z)

text: c

text: sinc(π(z

text: λ))

text: c

text: Im

text: λ

text: )e

text: 2π

text: Imλ

text: λ

text: λ

text: λ

text: π(z

text: λ)

text: λ

text: is

text: uniformly

text: bounded

text: from

text: below.

text: Indeed,

text: if

text: λ

text: is

text: such

text: that

text: Im

text: λ

text: then

text: λ))

text: eπ

text: λ

text: n

text: and

text: hence

text: sin(π(x

text: λ))

text: Im

text: λ

text: K

text: (x)

text: n−

text: C

text: λ

text: λ

text: π(xn

text: λ)

text: xn

text: λ

text: C

text: Thus

text: it

text: is

text: consider

text: points

text: λ

text: with

text: Im

text: λ

text: Let

text: x

text: be

text: point

text: of

text: S

text: closest

text: to

text: λ.

text: Then

text: there

text: is

text: δ

text: independent

text: of

text: λ,

text: such

text: that

text: sin(π(x

text: λ))

text: K

text: (x)

text: K

text: (x

text: δ.

text: λ

text: λ

text: n

text: C

text: λ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A SURVEY ON REVERSE CARLESON MEASURES 19 It is interesting to point out that µ is a Carleson measure for P W since S is in a strip and separated. ∈ T As was asked for the Paley-Wiener space P W , what are the µ M ( ) for which + (cid:107) (cid:107) (cid:16) (cid:107) (cid:107) ∀ ∈ K f f f ? m µ Θ In [54] Volberg generalized the pr(cid:90)evious results and gave a complete answer for general model ∈ ∞ T (cid:62) spaces and absolutely cont(cid:98)inuous measures dµ = wdm, where w L ( ), w 0. Let − | | 1 z 2 ∈ D w(z) = w(ζ)| −(cid:98) | dm(ζ), z , T z ζ 2 D be the Poisson integral of w and note that w is harmonic (and positive) on and has radial boundary values equal to w m-almost everywhere [17]. Theorem 6.17. Let dµ = wdm, with w ∈ L∞(T), w (cid:62) 0, and let Θ be an inner function. Then the following assertions are equivalent: (cid:107) (cid:107) (cid:16) (cid:107) (cid:107) ∈ K (i) f f for all f ; m µ Θ { } ⊂ D (cid:98) (ii) if λ (cid:62) , then n n 1 ⇒ | | (cid:98) l→im∞ w(λn) = 0 = l→im∞ Θ(λn) = 1; n n { | | ∈ D} (iii) inf w(λ) + Θ(λ) : λ > 0. In particular, this theorem applies to the special case when dµ = χ dm, with Σ a Borel subset of T Σ . However the conditions obtained from Volberg’s theorem are not expressed directly in terms of a density condition as was the case for P W (see Theorem 5.1). It is natural to ask if we can K obtain a characterization of dominating sets for in terms of a relative density. Dyakonov Θ answered this question in [19]. In the following result, H 2 is the Hardy space of the upper-half { } { } ⊥ plane Im z > 0 , Ψ is an inner function on Im z > 0 , and K = (ΨH 2) is a model space Ψ for the upper-half plane. { } Theorem 6.18. For an inner function Ψ on Im z > 0 the following are equivalent: (cid:48) ∈ ∞ R (i) Ψ L ( ); ⊂ R (ii) Every Lebesgue measurable set E for which these exists an δ > 0 and an η > 0 such that | ∩ − | (cid:62) ∀ ∈ R E [x η, x + η] δ x is dominating for the model space K . Ψ | (cid:48) | In the case corresponding to the Paley-Wiener space P W , Ψ(z) = e2iπz and thus Ψ (x) = 2π R (cid:48) ∈ ∞ R on . As was shown by Garnett [21], the condition Ψ L ( ) is equivalent to one of the following two conditions: ∃ (i) h > 0 such that {| | } inf Ψ(z) : 0 < Im(z) < h > 0; ##LTLine##

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

text: It

text: is

text: interesting

text: to

text: point

text: out

text: that

text: µ

text: is

text: a

text: Carleson

text: measure

text: for

text: P

text: W

text: since

text: S

text: is

text: in

text: a

text: strip

text: and

text: separated.

text: T

text: As

text: was

text: asked

text: for

text: the

text: Paley-Wiener

text: space

text: P

text: W

text: what

text: are

text: the

text: µ

text: M

text: for

text: which

text: K

text: f

text: f

text: f

text: m

text: µ

text: Θ

text: In

text: Volberg

text: generalized

text: the

text: results

text: and

text: gave

text: a

text: complete

text: answer

text: for

text: general

text: model

text: T

text: spaces

text: and

text: absolutely

text: measures

text: dµ

text: wdm,

text: where

text: w

text: L

text: w

text: Let

text: z

text: D

text: w(z)

text: w(ζ)|

text: dm(ζ),

text: z

text: T

text: z

text: ζ

text: D

text: be

text: the

text: Poisson

text: integral

text: of

text: w

text: and

text: note

text: that

text: w

text: is

text: harmonic

text: (and

text: positive)

text: on

text: and

text: has

text: radial

text: boundary

text: values

text: equal

text: to

text: w

text: m-almost

text: everywhere

text: Theorem

text: Let

text: dµ

text: wdm,

text: with

text: w

text: L∞(T),

text: w

text: and

text: let

text: Θ

text: be

text: an

text: inner

text: function.

text: Then

text: the

text: following

text: assertions

text: are

text: equivalent:

text: K

text: (i)

text: f

text: f

text: for

text: all

text: f

text: m

text: µ

text: Θ

text: D

text: (ii)

text: if

text: λ

text: then

text: n

text: n

text: l→im∞

text: w(λn)

text: l→im∞

text: Θ(λn)

text: n

text: n

text: D}

text: (iii)

text: inf

text: w(λ)

text: Θ(λ)

text: λ

text: In

text: particular,

text: this

text: theorem

text: applies

text: to

text: the

text: special

text: case

text: when

text: dµ

text: χ

text: dm,

text: with

text: Σ

text: a

text: Borel

text: subset

text: of

text: T

text: Σ

text: However

text: the

text: conditions

text: obtained

text: from

text: Volberg’s

text: theorem

text: are

text: not

text: expressed

text: directly

text: in

text: terms

text: of

text: a

text: density

text: condition

text: as

text: was

text: the

text: case

text: for

text: P

text: W

text: (see

text: Theorem

text: It

text: is

text: natural

text: to

text: ask

text: if

text: we

text: can

text: K

text: obtain

text: a

text: characterization

text: of

text: dominating

text: sets

text: for

text: in

text: terms

text: of

text: a

text: relative

text: density.

text: Dyakonov

text: Θ

text: answered

text: this

text: question

text: in

text: In

text: the

text: following

text: result,

text: H

text: is

text: the

text: Hardy

text: space

text: of

text: the

text: upper-half

text: plane

text: Im

text: z

text: Ψ

text: is

text: an

text: inner

text: function

text: on

text: Im

text: z

text: and

text: K

text: (ΨH

text: is

text: a

text: model

text: space

text: Ψ

text: for

text: the

text: upper-half

text: plane.

text: Theorem

text: For

text: an

text: inner

text: function

text: Ψ

text: on

text: Im

text: z

text: the

text: following

text: are

text: equivalent:

text: R

text: (i)

text: Ψ

text: L

text: R

text: (ii)

text: Every

text: Lebesgue

text: measurable

text: set

text: E

text: for

text: which

text: these

text: exists

text: an

text: δ

text: and

text: an

text: η

text: such

text: that

text: R

text: E

text: [x

text: η,

text: x

text: η]

text: δ

text: x

text: is

text: dominating

text: for

text: the

text: model

text: space

text: K

text: Ψ

text: In

text: the

text: case

text: corresponding

text: to

text: the

text: Paley-Wiener

text: space

text: P

text: W

text: Ψ(z)

text: and

text: thus

text: Ψ

text: (x)

text: 2π

text: R

text: R

text: on

text: As

text: was

text: shown

text: by

text: Garnett

text: the

text: condition

text: Ψ

text: L

text: is

text: equivalent

text: to

text: one

text: of

text: the

text: following

text: two

text: conditions:

text: (i)

text: h

text: such

text: that

text: inf

text: Ψ(z)

text: Im(z)

text: h

text: ##LTLine##


text: A SURVEY ON REVERSE CARLESON MEASURES 21 7. DE BRANGES-ROVNYAK SPACES These spaces are generalizations of the model spaces. Let ∞ { ∈ ∞ (cid:107) (cid:107) (cid:54) } H = f H : f ∞ 1 1 ∞ K be the closed unit ball in H . Recall that when Θ is inner, the model space is a closed Θ subspace of H2 with reproducing kernel function − 1 Θ(λ)Θ(z) ∈ D kΘ(z) = − , λ, z . λ 1 λz ∈ ∞ Using this as a guide, one can, for a given b H , deﬁne the de Branges-Rovnyak space H (b) 1 D to be the unique reproducing kernel Hilbert space of analytic functions on for which − 1 b(λ)b(z) ∈ D kb(z) = − , λ, z , λ 1 λz is the reproducing kernel [44]. Note(cid:88)that the function K(z, λ) := kb(z) is positive semi-deﬁnite D λ on , i.e., n (cid:62) a a K(λ , λ ) 0, i j i j i,j=1 { } D for all ﬁnite sets λ , . . . , λ of points in and all complex numbers a , . . . , a . Hence, we can 1 n 1 n associate to it a reproducing kernel Hilbert space and the above deﬁnition makes sense. There is an equivalent deﬁnition of H (b) via defects of certain Toeplitz operators [49]. It is well known that though these spaces play an important role in understanding contraction operators, the norms on these H (b) spaces, along with the elements contained in these spaces, (cid:107) (cid:107) ∞ remain mysterious. When b ∞ < 1 (i.e., b belongs to the interior of H ), then H (b) = H2 K1 with an equivalent norm. When b is an inner function, then H (b) = with the H2 norm. ∈ ∞ b For general b H , H (b) is contractively contained in H2 and this space is often called a 1 “sub-Hardy Hilbert space” [49]. The analysis of these H (b) spaces naturally splits into two ∞ distinct cases corresponding as to whether or not b is an extreme function for H , equivalently, − | | (cid:54)∈ 1 log(1 b ) L1(m). ∈ ∞ ∈ ∞ When b H is non-extreme, there is a unique outer function a H such that a(0) > 0 and 1 1 | | | | ∈ T (7.1) a(ξ) 2 + b(ξ) 2 = 1 m-a.e. ξ . Such a is often called the Pythagorean mate for b and the pair (a, b) is called a Pythagorean pair. There is the, now familiar, issue of boundary behavior of H (b) functions when deﬁning the (cid:107) (cid:107) integrals f in the Carleson and reverse Carleson testing conditions. With the model spaces µ (and with H2) there is a dense set of continuous functions for which one can sample in order (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) to test the Carleson ( f f ) and reverse Carleson conditions ( f f ). For a µ m m µ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

label: section

text: DE

text: BRANGES-ROVNYAK

text: SPACES

text: These

text: spaces

text: are

text: generalizations

text: of

text: the

text: model

text: spaces.

text: Let

text: H

text: f

text: H

text: f

text: K

text: be

text: the

text: closed

text: unit

text: ball

text: in

text: H

text: Recall

text: that

text: when

text: Θ

text: is

text: inner,

text: the

text: model

text: space

text: is

text: a

text: closed

text: Θ

text: subspace

text: of

text: with

text: reproducing

text: kernel

text: function

text: Θ(λ)Θ(z)

text: D

text: kΘ(z)

text: λ,

text: z

text: λ

text: λz

text: Using

text: this

text: as

text: a

text: guide,

text: one

text: can,

text: for

text: a

text: given

text: b

text: H

text: deﬁne

text: the

text: de

text: Branges-Rovnyak

text: space

text: H

text: (b)

text: D

text: to

text: be

text: the

text: unique

text: reproducing

text: kernel

text: Hilbert

text: space

text: of

text: analytic

text: functions

text: on

text: for

text: which

text: b(λ)b(z)

text: D

text: kb(z)

text: λ,

text: z

text: λ

text: λz

text: is

text: the

text: reproducing

text: kernel

text: the

text: function

text: K(z,

text: λ)

text: kb(z)

text: is

text: positive

text: semi-deﬁnite

text: D

text: λ

text: on

text: i.e.,

text: n

text: a

text: a

text: K(λ

text: λ

text: i

text: j

text: i

text: j

text: D

text: for

text: all

text: ﬁnite

text: sets

text: λ

text: λ

text: of

text: points

text: in

text: and

text: all

text: complex

text: numbers

text: a

text: a

text: Hence,

text: we

text: can

text: n

text: n

text: associate

text: to

text: it

text: a

text: reproducing

text: kernel

text: Hilbert

text: space

text: and

text: the

text: above

text: deﬁnition

text: makes

text: sense.

text: There

text: is

text: an

text: equivalent

text: deﬁnition

text: of

text: H

text: (b)

text: via

text: defects

text: of

text: certain

text: Toeplitz

text: operators

text: It

text: is

text: well

text: known

text: that

text: though

text: these

text: spaces

text: play

text: an

text: important

text: role

text: in

text: understanding

text: contraction

text: operators,

text: the

text: norms

text: on

text: these

text: H

text: (b)

text: spaces,

text: along

text: with

text: the

text: elements

text: contained

text: in

text: these

text: spaces,

text: remain

text: mysterious.

text: When

text: b

text: (i.e.,

text: b

text: belongs

text: to

text: the

text: interior

text: of

text: H

text: then

text: H

text: (b)

text: with

text: an

text: equivalent

text: norm.

text: When

text: b

text: is

text: an

text: inner

text: function,

text: then

text: H

text: (b)

text: with

text: the

text: norm.

text: b

text: For

text: general

text: b

text: H

text: H

text: (b)

text: is

text: contractively

text: contained

text: in

text: and

text: this

text: space

text: is

text: often

text: called

text: a

text: “sub-Hardy

text: Hilbert

text: space”

text: The

text: analysis

text: of

text: these

text: H

text: (b)

text: spaces

text: naturally

text: splits

text: into

text: two

text: distinct

text: cases

text: corresponding

text: as

text: to

text: whether

text: or

text: not

text: b

text: is

text: an

text: extreme

text: function

text: for

text: H

text: equivalently,

text: b

text: When

text: b

text: H

text: is

text: non-extreme,

text: there

text: is

text: a

text: unique

text: outer

text: function

text: a

text: H

text: such

text: that

text: and

text: T

text: a(ξ)

text: b(ξ)

text: m-a.e.

text: ξ

text: Such

text: a

text: is

text: often

text: called

text: the

text: Pythagorean

text: mate

text: for

text: b

text: and

text: the

text: pair

text: (a,

text: b)

text: is

text: called

text: a

text: Pythagorean

text: pair.

text: There

text: is

text: the,

text: now

text: familiar,

text: issue

text: of

text: boundary

text: behavior

text: of

text: H

text: (b)

text: functions

text: when

text: deﬁning

text: the

text: integrals

text: f

text: in

text: the

text: Carleson

text: and

text: reverse

text: Carleson

text: testing

text: conditions.

text: With

text: the

text: model

text: spaces

text: µ

text: (and

text: with

text: there

text: is

text: a

text: dense

text: set

text: of

text: continuous

text: functions

text: for

text: which

text: one

text: can

text: sample

text: in

text: order

text: to

text: test

text: the

text: Carleson

text: f

text: f

text: and

text: reverse

text: Carleson

text: conditions

text: f

text: f

text: For

text: a

text: µ

text: m

text: m

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 FRICAIN, HARTMANN, AND ROSS ∩ D− general H (b) space however, it is not quite clear whether or not H (b) C( ) is even non- zero. In certain circumstances, for example when b is non-extreme or when b is an inner function, ∩ D− H (b) C( ) is actually dense in H (b). For general extreme b, this remains unknown. Thus we are forced to make some deﬁnitions. Deﬁnition 7.2. For µ ∈ M (D−) we say that an analytic function f on D is µ-admissible if + T the non-tangential limits of f exist µ-almost everywhere on . We let H (b) denote the set of µ µ-admissible functions in H (b). ∈ | With this deﬁnition in mind, if f H (b) , then deﬁning f on the carrier of µ T via its non- (cid:107) (cid:107)µ ∞ tangential boundary values, we see that f is well deﬁned with a value in [0, + ]. µ D T Of course when µ is carried on , i.e., µ( ) = 0, then H (b) = H (b). Hence Deﬁnition 7.2 T µ only comes into play when µ has part of the unit circle in its carrier. Note that H (b) = H (b) ⊂ m since H (b) H2. However, there are often other µ, even ones with non-trivial singular parts on T with respect to m, for which H (b) = H (b) . The Clark measures associated with an inner µ function b have this property [8, 13]. Deﬁnition 7.3. A measure µ ∈ M (D−) is a Carleson measure for H (b) if H (b) = H (b) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ + µ and f f for all f H (b). µ b ≡ When b 0, i.e., when H (b) = H2 then, as a consequence of Carleson’s theorem (see Theorem (cid:46) | | | (cid:28) 2.2) for H2, we see that when µ satisﬁes µ(S ) I for all arcs I, then µ T m and so I H (b) = H (b). When b is an inner function, recall a discussion following (6.5) which says µ (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ ∩ D− that if the Carleson testing condition f f holds for all f H (b) C( ), then µ m H (b) = H (b). So in these two particular cases, the delicate issue of deﬁning the integrals in (cid:107) (cid:107) µ ∈ f for f H (b) seems to sort itself out. For general b, we do not have this luxury. µ Lacey et al. [30] solved the longstanding problem of characterizing the two-weight inequalities for Cauchy transforms. Let us take a moment to indicate how their results can be used to discuss Carleson measures for H (b). Let σ be th(cid:90)e Aleksandrov-Clark measure associated with b, that is ∈ T the unique σ M ( ) satisfying + − | | − | | 1 b(z) 2 1 z 2 ∈ D | − | = | − | dσ(ζ), z . 1 b(z) 2 T z ζ 2 −→ Let V : L2(σ) H (b) be the operat(cid:90)or deﬁned by b − f(ζ) − (7.4) (V f)(z) = (1 b(z)) − dσ(ζ) = (1 b(z))(C f)(z), b ¯ σ T 1 ζz (cid:90) where C is the Cauchy transform σ f(ζ) (C f)(z) = − dσ(ζ). σ T 1 ζz ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FRICAIN,

text: HARTMANN,

text: AND

text: ROSS

text: D−

text: general

text: H

text: (b)

text: space

text: however,

text: it

text: is

text: not

text: quite

text: clear

text: whether

text: or

text: not

text: H

text: (b)

text: C(

text: is

text: even

text: non-

text: zero.

text: In

text: certain

text: circumstances,

text: for

text: example

text: when

text: b

text: is

text: non-extreme

text: or

text: when

text: b

text: is

text: an

text: inner

text: function,

text: D−

text: H

text: (b)

text: C(

text: is

text: actually

text: dense

text: in

text: H

text: (b).

text: For

text: general

text: extreme

text: b,

text: this

text: remains

text: unknown.

text: Thus

text: we

text: are

text: forced

text: to

text: make

text: some

text: deﬁnitions.

text: Deﬁnition

text: For

text: µ

text: M

text: (D−)

text: we

text: say

text: that

text: an

text: analytic

text: function

text: f

text: on

text: D

text: is

text: µ-admissible

text: if

text: T

text: the

text: non-tangential

text: limits

text: of

text: f

text: exist

text: µ-almost

text: everywhere

text: on

text: We

text: let

text: H

text: (b)

text: denote

text: the

text: set

text: of

text: µ

text: µ-admissible

text: functions

text: in

text: H

text: (b).

text: With

text: this

text: deﬁnition

text: in

text: mind,

text: if

text: f

text: H

text: (b)

text: then

text: deﬁning

text: f

text: on

text: the

text: carrier

text: of

text: µ

text: T

text: via

text: its

text: non-

text: tangential

text: boundary

text: values,

text: we

text: see

text: that

text: f

text: is

text: well

text: deﬁned

text: with

text: a

text: value

text: in

text: µ

text: D

text: T

text: Of

text: course

text: when

text: µ

text: is

text: carried

text: on

text: i.e.,

text: µ(

text: then

text: H

text: (b)

text: H

text: (b).

text: Hence

text: Deﬁnition

text: T

text: µ

text: only

text: comes

text: into

text: play

text: when

text: µ

text: has

text: part

text: of

text: the

text: unit

text: circle

text: in

text: its

text: carrier.

text: Note

text: that

text: H

text: (b)

text: H

text: (b)

text: m

text: since

text: H

text: (b)

text: However,

text: there

text: are

text: often

text: other

text: µ,

text: even

text: ones

text: with

text: non-trivial

text: singular

text: parts

text: on

text: T

text: with

text: respect

text: to

text: m,

text: for

text: which

text: H

text: (b)

text: H

text: (b)

text: The

text: Clark

text: measures

text: associated

text: with

text: an

text: inner

text: µ

text: function

text: b

text: have

text: this

text: property

text: Deﬁnition

text: A

text: measure

text: µ

text: M

text: (D−)

text: is

text: a

text: Carleson

text: measure

text: for

text: H

text: (b)

text: if

text: H

text: (b)

text: H

text: (b)

text: µ

text: and

text: f

text: f

text: for

text: all

text: f

text: H

text: (b).

text: µ

text: b

text: When

text: b

text: i.e.,

text: when

text: H

text: (b)

text: then,

text: as

text: a

text: consequence

text: of

text: Carleson’s

text: theorem

text: (see

text: Theorem

text: for

text: we

text: see

text: that

text: when

text: µ

text: satisﬁes

text: µ(S

text: I

text: for

text: all

text: arcs

text: I,

text: then

text: µ

text: T

text: m

text: and

text: so

text: I

text: H

text: (b)

text: H

text: (b).

text: When

text: b

text: is

text: an

text: inner

text: function,

text: recall

text: a

text: discussion

text: following

text: which

text: says

text: µ

text: D−

text: that

text: if

text: the

text: Carleson

text: testing

text: condition

text: f

text: f

text: holds

text: for

text: all

text: f

text: H

text: (b)

text: C(

text: then

text: µ

text: m

text: H

text: (b)

text: H

text: (b).

text: So

text: in

text: these

text: two

text: particular

text: cases,

text: the

text: delicate

text: issue

text: of

text: deﬁning

text: the

text: integrals

text: in

text: µ

text: f

text: for

text: f

text: H

text: (b)

text: seems

text: to

text: sort

text: itself

text: out.

text: For

text: general

text: b,

text: we

text: do

text: not

text: have

text: this

text: luxury.

text: µ

text: Lacey

text: et

text: al.

text: solved

text: the

text: longstanding

text: problem

text: of

text: characterizing

text: the

text: two-weight

text: inequalities

text: for

text: Cauchy

text: transforms.

text: Let

text: us

text: take

text: a

text: moment

text: to

text: indicate

text: how

text: their

text: results

text: can

text: be

text: used

text: to

text: discuss

text: Carleson

text: measures

text: for

text: H

text: (b).

text: Let

text: σ

text: be

text: Aleksandrov-Clark

text: measure

text: associated

text: with

text: b,

text: that

text: is

text: T

text: the

text: unique

text: σ

text: M

text: satisfying

text: b(z)

text: z

text: D

text: dσ(ζ),

text: z

text: b(z)

text: T

text: z

text: ζ

text: Let

text: V

text: H

text: (b)

text: be

text: the

text: deﬁned

text: by

text: b

text: f(ζ)

text: (V

text: f)(z)

text: b(z))

text: dσ(ζ)

text: b(z))(C

text: f)(z),

text: b

text: σ

text: T

text: ζz

text: where

text: C

text: is

text: the

text: Cauchy

text: transform

text: σ

text: f(ζ)

text: (C

text: f)(z)

text: dσ(ζ).

text: σ

text: T

text: ζz

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##