text: Among the ﬁrst fundamental and strikingly new phenomenon in complex dimensions N any greater than one, is the Hartogs phenomenon: every holomorphic function on a punctured poly- disc extends across the puncture, so that in particular, there are no isolated singularities for holomorphic functions on domains in dimensions N > 1. For convenience in sketching a quick \ { } proof, let us demonstrate this phenomenon on Up := U p where U is the polydisc centered C at the origin in 2 of polyradius (2, 2) with the puncture p = (1, 1). Given any holomorphic function f on Up, we apply the fact mentioned in the foregoing para to the restriction of f to the complete Reinhardt domain L, where L is the subdomain of Up whose absolute diagram was sketched in ﬁgure 1; the aforementioned fact ensures a representation of f by a single power series compactly convergent (at least) on L. But then as L fails to be logarithmically convex, we conclude that this power series must converge on some neighbourhood of p as well. As the func- tion deﬁned by this power series already agrees with f on the open set L, the identity principle guarantees that this is indeed a holomorphic extension of f across p. This ﬁnishes the proof that all functions holomorphic on the domain Up extend ‘simultaneously’ across the boundary point p. It is then natural to single out domains maximal with respect to this phenomenon of simulta- neous extension of holomorphic functions i.e., domains D such that for each boundary point ∈ p ∂D, there is a function f holomorphic on D resisting holomorphic continuation to any p neighbourhood of p. A domain possessing this property is called a domain of holomorphy. It turns out that this property is equivalent to the stronger requirement that there exist at least one holomorphic function which does not extend holomorphically across the boundary near any point in ∂D. Infact this is only one among multiple equivalent deﬁnitions/characterizations of domains of holomorphy. A celebrated problem going by the name of the Levi problem and taking several decades for its complete resolution, was to obtain a geometric characterization of domains of holomorphy. This is best left for another essay; suﬃce it to say here that the answer lies in a subtle convexity property and we refer the reader again to [7], [8] and other texts of the subject. Our next goal here will be to show that domains of convergence of power series are indeed domains of holomorphy. The question to be dealt with now is: given a domain D which is the domain of convergence of some power series (equivalently, a logarithmically convex complete Reinhardt domain D) in C N and an arbitrary point p of its boundary ∂D, is it possible to construct (another) power → ∞ series f (z) which converges on D and whose limit supremum as z p is ? Note that this p question does not get trivially settled with the knowledge of the existence of a power series converging precisely on D, owing to the possibility of the existence some (tiny) piece of ∂D across which all such power series can somehow be continued holomorphically. As already seen at (2.22), while constructing power series with certain desired properties, it is best to use the freedom in expressing them as a sum of monomials in any order that we wish – in a manner that is telling about the desired properties. With this ﬂexibility, let us demonstrate that domains of convergence of power series are (what are known as ‘weak’-) domains of holomorphy by constructing the function f in question. We cannot help but narrate here the clear but concise p treatment in Ohsawa’s little text [18]. Suppose ﬁrst that D is bounded and observe that given ∈ C \ any point p in the exterior of D (i.e., p N D), there exists a monomial m (z) such that p | | (2.25) sup m (z) < m (p) = 1. ∈ p p z D Indeed, this follows essentially by passing to the logarithmic image G = λ(D), applying to it a standard separation theorem to the convex domain G and then exponentiating back – the only possible hitch in this process arising when some of the coeﬃcients of the gradient vector of 29 ##LTLine##

text: Among

text: the

text: ﬁrst

text: fundamental

text: and

text: strikingly

text: new

text: phenomenon

text: in

text: complex

text: dimensions

text: N

text: any

text: greater

text: than

text: one,

text: is

text: the

text: Hartogs

text: phenomenon:

text: every

text: holomorphic

text: function

text: on

text: a

text: punctured

text: poly-

text: disc

text: extends

text: across

text: the

text: puncture,

text: so

text: that

text: in

text: particular,

text: there

text: are

text: no

text: isolated

text: singularities

text: for

text: holomorphic

text: functions

text: on

text: domains

text: in

text: dimensions

text: N

text: For

text: convenience

text: in

text: sketching

text: a

text: quick

text: proof,

text: let

text: us

text: demonstrate

text: this

text: phenomenon

text: on

text: Up

text: U

text: p

text: where

text: U

text: is

text: the

text: polydisc

text: centered

text: C

text: at

text: the

text: origin

text: in

text: of

text: polyradius

text: with

text: the

text: puncture

text: p

text: Given

text: any

text: holomorphic

text: function

text: f

text: on

text: Up,

text: we

text: apply

text: the

text: fact

text: mentioned

text: in

text: the

text: foregoing

text: para

text: to

text: the

text: restriction

text: of

text: f

text: to

text: the

text: complete

text: Reinhardt

text: domain

text: L,

text: where

text: L

text: is

text: the

text: subdomain

text: of

text: Up

text: whose

text: absolute

text: diagram

text: was

text: sketched

text: in

text: ﬁgure

text: the

text: aforementioned

text: fact

text: ensures

text: a

text: representation

text: of

text: f

text: by

text: a

text: single

text: power

text: series

text: compactly

text: convergent

text: (at

text: least)

text: on

text: L.

text: But

text: then

text: as

text: L

text: fails

text: to

text: be

text: logarithmically

text: convex,

text: we

text: conclude

text: that

text: this

text: power

text: series

text: must

text: converge

text: on

text: some

text: neighbourhood

text: of

text: p

text: as

text: well.

text: As

text: the

text: func-

text: tion

text: deﬁned

text: by

text: this

text: power

text: series

text: already

text: agrees

text: with

text: f

text: on

text: the

text: open

text: set

text: L,

text: the

text: identity

text: principle

text: guarantees

text: that

text: this

text: is

text: indeed

text: a

text: holomorphic

text: extension

text: of

text: f

text: across

text: p.

text: This

text: ﬁnishes

text: the

text: proof

text: that

text: all

text: functions

text: holomorphic

text: on

text: the

text: domain

text: Up

text: extend

text: ‘simultaneously’

text: across

text: the

text: boundary

text: point

text: p.

text: It

text: is

text: then

text: natural

text: to

text: single

text: out

text: domains

text: maximal

text: with

text: respect

text: to

text: this

text: phenomenon

text: of

text: simulta-

text: neous

text: extension

text: of

text: holomorphic

text: functions

text: i.e.,

text: domains

text: D

text: such

text: that

text: for

text: each

text: boundary

text: point

text: p

text: ∂D,

text: there

text: is

text: a

text: function

text: f

text: holomorphic

text: on

text: D

text: resisting

text: holomorphic

text: continuation

text: to

text: any

text: p

text: neighbourhood

text: of

text: p.

text: A

text: domain

text: possessing

text: this

text: property

text: is

text: called

text: a

text: domain

text: of

text: holomorphy.

text: It

text: turns

text: out

text: that

text: this

text: property

text: is

text: equivalent

text: to

text: the

text: stronger

text: requirement

text: that

text: there

text: exist

text: at

text: least

text: one

text: holomorphic

text: function

text: which

text: does

text: not

text: extend

text: holomorphically

text: across

text: the

text: boundary

text: near

text: any

text: point

text: in

text: ∂D.

text: Infact

text: this

text: is

text: only

text: one

text: among

text: multiple

text: equivalent

text: deﬁnitions/characterizations

text: of

text: domains

text: of

text: holomorphy.

text: A

text: celebrated

text: problem

text: going

text: by

text: the

text: name

text: of

text: the

text: Levi

text: problem

text: and

text: taking

text: several

text: decades

text: for

text: its

text: complete

text: resolution,

text: was

text: to

text: obtain

text: a

text: geometric

text: characterization

text: of

text: domains

text: of

text: holomorphy.

text: This

text: is

text: best

text: left

text: for

text: another

text: essay;

text: suﬃce

text: it

text: to

text: say

text: here

text: that

text: the

text: answer

text: lies

text: in

text: a

text: subtle

text: convexity

text: property

text: and

text: we

text: refer

text: the

text: reader

text: again

text: to

text: and

text: other

text: texts

text: of

text: the

text: subject.

text: Our

text: next

text: goal

text: here

text: will

text: be

text: to

text: show

text: that

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: are

text: indeed

text: domains

text: of

text: holomorphy.

text: The

text: question

text: to

text: be

text: dealt

text: with

text: now

text: is:

text: given

text: a

text: domain

text: D

text: which

text: is

text: the

text: domain

text: of

text: convergence

text: of

text: some

text: power

text: series

text: (equivalently,

text: a

text: logarithmically

text: convex

text: complete

text: Reinhardt

text: domain

text: D)

text: in

text: C

text: N

text: and

text: an

text: arbitrary

text: point

text: p

text: of

text: its

text: boundary

text: ∂D,

text: is

text: it

text: possible

text: to

text: construct

text: (another)

text: power

text: series

text: f

text: (z)

text: which

text: converges

text: on

text: D

text: and

text: whose

text: limit

text: supremum

text: as

text: z

text: p

text: is

text: Note

text: that

text: this

text: p

text: question

text: does

text: not

text: get

text: trivially

text: settled

text: with

text: the

text: knowledge

text: of

text: the

text: existence

text: of

text: a

text: power

text: series

text: converging

text: precisely

text: on

text: D,

text: owing

text: to

text: the

text: possibility

text: of

text: the

text: existence

text: some

text: (tiny)

text: piece

text: of

text: ∂D

text: across

text: which

text: all

text: such

text: power

text: series

text: can

text: somehow

text: be

text: continued

text: holomorphically.

text: As

text: already

text: seen

text: at

text: while

text: constructing

text: power

text: series

text: with

text: certain

text: desired

text: properties,

text: it

text: is

text: best

text: to

text: use

text: the

text: freedom

text: in

text: expressing

text: them

text: as

text: a

text: sum

text: of

text: monomials

text: in

text: any

text: order

text: that

text: we

text: wish

text: in

text: a

text: manner

text: that

text: is

text: telling

text: about

text: the

text: desired

text: properties.

text: With

text: this

text: ﬂexibility,

text: let

text: us

text: demonstrate

text: that

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: are

text: (what

text: are

text: known

text: as

text: ‘weak’-)

text: domains

text: of

text: holomorphy

text: by

text: constructing

text: the

text: function

text: f

text: in

text: question.

text: We

text: cannot

text: help

text: but

text: narrate

text: here

text: the

text: clear

text: but

text: concise

text: p

text: treatment

text: in

text: Ohsawa’s

text: little

text: text

text: Suppose

text: ﬁrst

text: that

text: D

text: is

text: bounded

text: and

text: observe

text: that

text: given

text: C

text: any

text: point

text: p

text: in

text: the

text: exterior

text: of

text: D

text: (i.e.,

text: p

text: N

text: D),

text: there

text: exists

text: a

text: monomial

text: m

text: (z)

text: such

text: that

text: p

text: sup

text: m

text: (z)

text: m

text: (p)

text: p

text: p

text: z

text: D

text: Indeed,

text: this

text: follows

text: essentially

text: by

text: passing

text: to

text: the

text: logarithmic

text: image

text: G

text: λ(D),

text: applying

text: to

text: it

text: a

text: standard

text: separation

text: theorem

text: to

text: the

text: convex

text: domain

text: G

text: and

text: then

text: exponentiating

text: back

text: the

text: only

text: possible

text: hitch

text: in

text: this

text: process

text: arising

text: when

text: some

text: of

text: the

text: coeﬃcients

text: of

text: the

text: gradient

text: vector

text: of

text: ##LTLine##


text: C −∞ ∞ T with N. Products of domains in [ , )N with N are pushed forward by this mapping A T C onto domains which are ‘multi-circular’ (invariant under the natural action of N on N) and R are termed Reinhardt domains 8. Pull-backs of convex domains in N by λ are called logarith- mically convex – formulated again precisely in deﬁnition (2.11) below. So, sets closed under Ψ are those whose logarithmic images are closed under the former binary operation Φ . As t t Φ requires no coordinatization, it is trivial that sets closed under this binary operation for all ∈t t I namely the convex sets, remain convex under all aﬃne transformations – convexity is an aﬃne property. However, it is far more non-trivial that multi-circular logarithmically convex C domains in N whose logarithmic images are complete/closed under translation by vectors from −R ( +)N, possess a property which remains invariant under all biholomorphic (not just aﬃne!) transformations. This property known as pseudoconvexity will not be discussed much here (we refer the novice to Range’s expository articles [7] and [8]). Pseudoconvexity is a subtle property; however, we hope that the present essay, among other extensive treatises such as [11], convinces the reader that it is possible to gain a ‘hands-on’ experience with the simplest examples of ‘pseu- doconvex’ domains namely, domains of convergence of power series in several complex variables. (cid:80) Among the most elementary functions of several complex variables are the monomial functions and their linear combinations. Deﬁnition 1.1. A function of the form p(z) = | |≤ c zJ is called a polynomial. Here, if at | | J m J least one of the c ’s with J = m is non-zero, the total degree of p is deﬁned to be deg(p) = m. J For the zero polynomial, the degree is not deﬁned. A polynomial is called homogeneous (of | | degree m) if the coeﬃcients c for J < m are all zero. Equivalently, a polynomial p of degree J ∈ C m is homogeneous if and only if p(λz) = λmp(z) for all λ . C Thus polynomials are for us by deﬁnition, functions on the coordinate space N, deﬁned by C C expressions from the (coordinate) ring [z] = [z , . . . , z ]. Such functions are annihilated 1 N by the operators ∂/∂z for all j = 1, . . . , N and are sometimes referred to as ‘holomorphic j polynomials’ to distinguish them from ﬁnite linear combinations of monomials in the 2N many ‘independent’ variables (z, z) = (z , . . . , z , z , . . . , z ). A basic question arising here is of 1 N 1 N the ‘independence’ of z from z which is addressed in basic complex analysis; for an advanced, enlightening treatment we refer the reader to [5]. We shall not dwell anymore on this than saying that each z is annihilated by the operator ∂/∂z for instance, where we request the j j reader to recall the notion of Wirtinger derivatives here: for example ∂/∂z is the complex linear j combination of the standard partial diﬀerential operators ∂/∂x , ∂/∂y given by 1/2 (∂/∂x + (cid:60) (cid:61) j j j i∂/∂y ) where x = z , y = z . Moving further, we may obtain more functions by taking j j j j j C limits of polynomials; but such limits will often not be well-deﬁned on all of N and we need to identify the subset on which they exist. Before we investigate this, we must ﬁrst be clear about issues of limits and convergence in several variables, which we review in the following (cid:80) sub-section. ∈ N Suppose that for each J N, a complex number c is given; we may form the series c and 0 J J discuss the matter of its convergence. A trouble immediately arising is: there is no canonical N order on N. So to start with, we make the follow(cid:80)ing 0 (cid:80) ∈ N Deﬁnition 1.2. The series of complex numbers c indexed by J N is said to be conver- N → NJ ∞ 0| | ∞ gent, if there exists at least one bijection φ : N such that c < . Then the 0 i=1 φ(i) ≤ 8It is helpful to draw (for N 3) images of Reinhardt domains in the absolute space as well as in the corresponding logarithmic space and we urge the reader to do so. 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C

label: paragraph

text: T

text: with

text: N.

text: Products

text: of

text: domains

text: in

text: )N

text: with

text: N

text: are

text: pushed

text: forward

text: by

text: this

text: mapping

text: A

text: T

text: C

text: onto

text: domains

text: which

text: are

text: ‘multi-circular’

text: (invariant

text: under

text: the

text: natural

text: action

text: of

text: N

text: on

text: N)

text: and

text: R

text: are

text: termed

text: Reinhardt

text: domains

text: Pull-backs

text: of

text: convex

text: domains

text: in

text: N

text: by

text: λ

text: are

text: called

text: logarith-

text: mically

text: convex

text: formulated

text: again

text: precisely

text: in

text: deﬁnition

text: below.

text: So,

text: sets

text: closed

text: under

text: Ψ

text: are

text: those

text: whose

text: logarithmic

text: images

text: are

text: closed

text: under

text: the

text: former

text: binary

text: operation

text: Φ

text: As

text: t

text: t

text: Φ

text: requires

text: no

text: coordinatization,

text: it

text: is

text: trivial

text: that

text: sets

text: closed

text: under

text: this

text: binary

text: operation

text: for

text: all

text: ∈t

text: t

text: I

text: namely

text: the

text: convex

text: sets,

text: remain

text: convex

text: under

text: all

text: aﬃne

text: transformations

text: convexity

text: is

text: an

text: aﬃne

text: property.

text: However,

text: it

text: is

text: far

text: more

text: non-trivial

text: that

text: multi-circular

text: logarithmically

text: convex

text: C

text: domains

text: in

text: N

text: whose

text: logarithmic

text: images

text: are

text: complete/closed

text: under

text: translation

text: by

text: vectors

text: from

text: −R

text: +)N,

text: possess

text: a

text: property

text: which

text: remains

text: invariant

text: under

text: all

text: biholomorphic

text: (not

text: just

text: aﬃne!)

text: transformations.

text: This

text: property

text: known

text: as

text: pseudoconvexity

text: will

text: not

text: be

text: discussed

text: much

text: here

text: (we

text: refer

text: the

text: novice

text: to

text: Range’s

text: expository

text: articles

text: and

text: Pseudoconvexity

text: is

text: a

text: subtle

text: property;

text: however,

text: we

text: hope

text: that

text: the

text: present

text: essay,

text: among

text: other

text: extensive

text: treatises

text: such

text: as

text: convinces

text: the

text: reader

text: that

text: it

text: is

text: possible

text: to

text: gain

text: a

text: ‘hands-on’

text: experience

text: with

text: the

text: simplest

text: examples

text: of

text: ‘pseu-

text: doconvex’

text: domains

text: namely,

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: in

text: several

text: complex

text: variables.

text: Among

text: the

text: most

text: elementary

text: functions

text: of

text: several

text: complex

text: variables

text: are

text: the

text: monomial

text: functions

text: and

text: their

text: linear

text: combinations.

text: Deﬁnition

text: A

text: function

text: of

text: the

text: form

text: p(z)

text: c

text: zJ

text: is

text: called

text: a

text: polynomial.

text: Here,

text: if

text: at

text: J

text: m

text: J

text: least

text: one

text: of

text: the

text: c

text: ’s

text: with

text: J

text: m

text: is

text: non-zero,

text: the

text: total

text: degree

text: of

text: p

text: is

text: deﬁned

text: to

text: be

text: deg(p)

text: m.

text: J

text: For

text: the

text: zero

text: polynomial,

text: the

text: degree

text: is

text: not

text: deﬁned.

text: A

text: polynomial

text: is

text: called

text: homogeneous

text: (of

text: degree

text: m)

text: if

text: the

text: coeﬃcients

text: c

text: for

text: J

text: m

text: are

text: all

text: zero.

text: Equivalently,

text: a

text: polynomial

text: p

text: of

text: degree

text: J

text: C

text: m

text: is

text: homogeneous

text: if

text: and

text: only

text: if

text: p(λz)

text: λmp(z)

text: for

text: all

text: λ

text: C

text: Thus

text: polynomials

text: are

text: for

text: us

text: by

text: deﬁnition,

text: functions

text: on

text: the

text: coordinate

text: space

text: N,

text: deﬁned

text: by

text: C

text: C

text: expressions

text: from

text: the

text: (coordinate)

text: ring

text: [z]

text: [z

text: z

text: Such

text: functions

text: are

text: annihilated

text: N

text: by

text: the

text: operators

text: ∂/∂z

text: for

text: all

text: j

text: N

text: and

text: are

text: sometimes

text: referred

text: to

text: as

text: ‘holomorphic

text: j

text: polynomials’

text: to

text: distinguish

text: them

text: from

text: ﬁnite

text: linear

text: combinations

text: of

text: monomials

text: in

text: the

text: many

text: ‘independent’

text: variables

text: (z,

text: z)

text: (z

text: z

text: z

text: z

text: A

text: basic

text: question

text: arising

text: here

text: is

text: of

text: N

text: N

text: the

text: ‘independence’

text: of

text: z

text: from

text: z

text: which

text: is

text: addressed

text: in

text: basic

text: complex

text: analysis;

text: for

text: an

text: advanced,

text: enlightening

text: treatment

text: we

text: refer

text: the

text: reader

text: to

text: We

text: shall

text: not

text: dwell

text: anymore

text: on

text: this

text: than

text: saying

text: that

text: each

text: z

text: is

text: annihilated

text: by

text: the

text: operator

text: ∂/∂z

text: for

text: instance,

text: where

text: we

text: request

text: the

text: j

text: j

text: reader

text: to

text: recall

text: the

text: notion

text: of

text: Wirtinger

text: derivatives

text: here:

text: for

text: example

text: ∂/∂z

text: is

text: the

text: complex

text: linear

text: j

text: combination

text: of

text: the

text: standard

text: partial

text: diﬀerential

text: operators

text: ∂/∂x

text: ∂/∂y

text: given

text: by

text: (∂/∂x

text: j

text: j

text: j

text: i∂/∂y

text: where

text: x

text: z

text: y

text: z

text: Moving

text: further,

text: we

text: may

text: obtain

text: more

text: functions

text: by

text: taking

text: j

text: j

text: j

text: j

text: j

text: C

text: limits

text: of

text: polynomials;

text: but

text: such

text: limits

text: will

text: often

text: not

text: be

text: well-deﬁned

text: on

text: all

text: of

text: N

text: and

text: we

text: need

text: to

text: identify

text: the

text: subset

text: on

text: which

text: they

text: exist.

text: Before

text: we

text: investigate

text: this,

text: we

text: must

text: ﬁrst

text: be

text: clear

text: about

text: issues

text: of

text: limits

text: and

text: convergence

text: in

text: several

text: variables,

text: which

text: we

text: review

text: in

text: the

text: following

text: sub-section.

text: N

text: Suppose

text: that

text: for

text: each

text: J

text: N,

text: a

text: complex

text: number

text: c

text: is

text: given;

text: we

text: may

text: form

text: the

text: series

text: c

text: and

text: J

text: J

text: discuss

text: the

text: matter

text: of

text: its

text: convergence.

text: A

text: trouble

text: immediately

text: arising

text: is:

text: there

text: is

text: no

text: canonical

text: N

text: order

text: on

text: N.

text: So

text: to

text: start

text: with,

text: we

text: make

text: the

text: N

text: Deﬁnition

text: The

text: series

text: of

text: complex

text: numbers

text: c

text: indexed

text: by

text: J

text: N

text: is

text: said

text: to

text: be

text: conver-

text: N

text: NJ

text: gent,

text: if

text: there

text: exists

text: at

text: least

text: one

text: bijection

text: φ

text: N

text: such

text: that

text: c

text: Then

text: the

text: φ(i)

text: is

text: helpful

text: to

text: draw

text: (for

text: N

text: images

text: of

text: Reinhardt

text: domains

text: in

text: the

text: absolute

text: space

text: as

text: well

text: as

text: in

text: the

text: corresponding

text: logarithmic

text: space

text: and

text: we

text: urge

text: the

text: reader

text: to

text: do

text: so.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: −∞ −∞ inﬁnite box-neighbourhood of ( , . . . , ) (at the ‘left-bottom’) arising as the logarithmic image of the polydisc spanned by any point is contained in G and consequently in all the G ’s · j as well owing to the concavity of the function dist( , ∂G) on G; this ensures that all the D ’s j are complete Reinhardt domains as well. If z, w are a pair of points in G whose distance from · ∂G are at least δ, then concavity of the function dist( , ∂G) on G, ensures that the minimum distance of every point of the line segment joining z, w in G lies at a distance at least δ from ∂G. This fact ensures that all the domains G ’s are convex and thereby the logarithmic convexity j of the D ’s. Thus, the D ’s form an (increasing) exhaustion of D by bounded logarithmically j j | | ∞ convex complete Reinhardt domains. Now, to construct an f with lim sup f (z) = , what p → p z p could be more simple than to arrange for a function whose values at some sequence p of points j in D approaching p, is at least as big as j? In trying to arrange for such a function f , we must p not loose sight of the requirement that f is to be given by a power series which converges on all p of D. Recall the availability of a characterizing test to determine whether or not a point belongs to the domain of convergence of any given power series, namely proposition 2.5. Put in words, according to this proposition, a point is within the domain of convergence of a power series if the sequence of complex numbers obtained by evaluating the monomials constituting the power series (in the standard partial ordering by degree) at that point, decays to zero at least at an exponential rate; stated diﬀerently, faster than a geometric progression (of ratio < 1). The last statement holds with the word ‘point’ replaced by ‘any point from the set of all points whose distance to the boundary of the domain of convergence is bounded below by a positive constant’. { } We choose the standard geometric progression namely 1/2k for measuring/controlling the rate in what follows. First, let pj be sequence in D which converges to p; indeed, choose the sequence ∈ \ so that (it escapes out of the D ’s linearly as:) pj D D and converges to p. Corresponding j j+1 j to each such pj, by (2.25) choose a monomial m whose value at pj exceeds the supremum of pj its values on D . We wish to arrange our series f in such a way that the value of the n-th term j p of the series, at pn, exceeds n – the amount by which it exceeds, is arranged to cancel out the − possible negative contributions of the remaining terms, so as to ensure (f (n) > n 1) ultimately → ∞ p that f (pn) . For instance, we may take the n-th term to be c m (z) with c > n, whose p n pn n value at pn is c . The major part of the ‘negative contributions’ to possibly pull down the value n of f at pn, will conceivably due to the terms preceding the n-th term, as the remaining tail p of the series f (assuming convergence) will be small. Put together with the aforementioned p ∈ N convergence criterion, we are then led to s(cid:12)(cid:12)e(cid:88)ek for(cid:0) sequenc(cid:1)es (cid:12)(cid:12)nk and real numbers ck such that (cid:12) (cid:12) − k 1 c = k + c m (p ) nj k j pj k (cid:12) (cid:12) (cid:12) j=1 (cid:12) together with the requirement sup c (m (z))nk < 1/2k. ∈ k pk z D k It is easy to construct the sequences c (cid:88)and (cid:0)n indu(cid:1)ctively, satisfying the above conditions at k k each stage. Then the series ∞ c m (z) nj j pj j=1 is compactly convergent (recall D ’s are relatively compact) on D and thus deﬁnes a holomor- j − ∞ phic function f (z) on D. As f (pn) > n 1, we must have lim sup f (z) = , with which we p p → p z p have attained our goal of checking out that domains of convergence of power series are indeed domains of holomorphy. 31

text: inﬁnite

text: box-neighbourhood

text: of

text: (at

text: the

text: ‘left-bottom’)

text: arising

text: as

text: the

text: logarithmic

text: image

text: of

text: the

text: polydisc

text: spanned

text: by

text: any

text: point

text: is

text: contained

text: in

text: G

text: and

text: consequently

text: in

text: all

text: the

text: G

text: ’s

text: j

text: as

text: well

text: owing

text: to

text: the

text: concavity

text: of

text: the

text: function

text: dist(

text: ∂G)

text: on

text: G;

text: this

text: ensures

text: that

text: all

text: the

text: D

text: ’s

text: j

text: are

text: complete

text: Reinhardt

text: domains

text: as

text: well.

text: If

text: z,

text: w

text: are

text: a

text: pair

text: of

text: points

text: in

text: G

text: whose

text: distance

text: from

text: ∂G

text: are

text: at

text: least

text: δ,

text: then

text: concavity

text: of

text: the

text: function

text: dist(

text: ∂G)

text: on

text: G,

text: ensures

text: that

text: the

text: minimum

text: distance

text: of

text: every

text: point

text: of

text: the

text: line

text: segment

text: joining

text: z,

text: w

text: in

text: G

text: lies

text: at

text: a

text: distance

text: at

text: least

text: δ

text: from

text: ∂G.

text: This

text: fact

text: ensures

text: that

text: all

text: the

text: domains

text: G

text: ’s

text: are

text: convex

text: and

text: thereby

text: the

text: logarithmic

text: convexity

text: j

text: of

text: the

text: D

text: ’s.

text: Thus,

text: the

text: D

text: ’s

text: form

text: an

text: (increasing)

text: exhaustion

text: of

text: D

text: by

text: bounded

text: logarithmically

text: j

text: j

text: convex

text: complete

text: Reinhardt

text: domains.

text: Now,

text: to

text: construct

text: an

text: f

text: with

text: lim

text: sup

text: f

text: (z)

text: what

text: p

text: p

text: z

text: p

text: could

text: be

text: more

text: simple

text: than

text: to

text: arrange

text: for

text: a

text: function

text: whose

text: values

text: at

text: some

text: sequence

text: p

text: of

text: points

text: j

text: in

text: D

text: approaching

text: p,

text: is

text: at

text: least

text: as

text: big

text: as

text: j?

text: In

text: trying

text: to

text: arrange

text: for

text: such

text: a

text: function

text: f

text: we

text: must

text: p

text: not

text: loose

text: sight

text: of

text: the

text: requirement

text: that

text: f

text: is

text: to

text: be

text: given

text: by

text: a

text: power

text: series

text: which

text: converges

text: on

text: all

text: p

text: of

text: D.

text: Recall

text: the

text: availability

text: of

text: a

text: characterizing

text: test

text: to

text: determine

text: whether

text: or

text: not

text: a

text: point

text: belongs

text: to

text: the

text: domain

text: of

text: convergence

text: of

text: any

text: given

text: power

text: series,

text: namely

text: proposition

text: Put

text: in

text: words,

text: according

text: to

text: this

text: proposition,

text: a

text: point

text: is

text: within

text: the

text: domain

text: of

text: convergence

text: of

text: a

text: power

text: series

text: if

text: the

text: sequence

text: of

text: complex

text: numbers

text: obtained

text: by

text: evaluating

text: the

text: monomials

text: constituting

text: the

text: power

text: series

text: (in

text: the

text: standard

text: partial

text: ordering

text: by

text: degree)

text: at

text: that

text: point,

text: decays

text: to

text: zero

text: at

text: least

text: at

text: an

text: exponential

text: rate;

text: stated

text: diﬀerently,

text: faster

text: than

text: a

text: geometric

text: progression

text: (of

text: ratio

text: The

text: last

text: statement

text: holds

text: with

text: the

text: word

text: ‘point’

text: replaced

text: by

text: ‘any

text: point

text: from

text: the

text: set

text: of

text: all

text: points

text: whose

text: distance

text: to

text: the

text: boundary

text: of

text: the

text: domain

text: of

text: convergence

text: is

text: bounded

text: below

text: by

text: a

text: positive

text: constant’.

text: We

text: choose

text: the

text: standard

text: geometric

text: progression

text: namely

text: for

text: measuring/controlling

text: the

text: rate

text: in

text: what

text: follows.

text: First,

text: let

text: pj

text: be

text: sequence

text: in

text: D

text: which

text: converges

text: to

text: p;

text: indeed,

text: choose

text: the

text: sequence

text: so

text: that

text: (it

text: escapes

text: out

text: of

text: the

text: D

text: ’s

text: linearly

text: as:)

text: pj

text: D

text: D

text: and

text: converges

text: to

text: p.

text: Corresponding

text: j

text: j

text: to

text: each

text: such

text: pj,

text: by

text: choose

text: a

text: monomial

text: m

text: whose

text: value

text: at

text: pj

text: exceeds

text: the

text: supremum

text: of

text: pj

text: its

text: values

text: on

text: D

text: We

text: wish

text: to

text: arrange

text: our

text: series

text: f

text: in

text: such

text: a

text: way

text: that

text: the

text: value

text: of

text: the

text: n-th

text: term

text: j

text: p

text: of

text: the

text: series,

text: at

text: pn,

text: exceeds

text: n

text: the

text: amount

text: by

text: which

text: it

text: exceeds,

text: is

text: arranged

text: to

text: cancel

text: out

text: the

text: possible

text: negative

text: contributions

text: of

text: the

text: remaining

text: terms,

text: so

text: as

text: to

text: ensure

text: (f

text: (n)

text: n

text: ultimately

text: p

text: that

text: f

text: (pn)

text: For

text: instance,

text: we

text: may

text: take

text: the

text: n-th

text: term

text: to

text: be

text: c

text: m

text: (z)

text: with

text: c

text: n,

text: whose

text: p

text: n

text: pn

text: n

text: value

text: at

text: pn

text: is

text: c

text: The

text: major

text: part

text: of

text: the

text: ‘negative

text: contributions’

text: to

text: possibly

text: pull

text: down

text: the

text: value

text: n

text: of

text: f

text: at

text: pn,

text: will

text: conceivably

text: due

text: to

text: the

text: terms

text: preceding

text: the

text: n-th

text: term,

text: as

text: the

text: remaining

text: tail

text: p

text: of

text: the

text: series

text: f

text: (assuming

text: convergence)

text: will

text: be

text: small.

text: Put

text: together

text: with

text: the

text: aforementioned

text: p

text: N

text: convergence

text: criterion,

text: we

text: are

text: then

text: led

text: to

text: and

text: real

text: numbers

text: ck

text: such

text: that

text: k

text: c

text: k

text: c

text: m

text: (p

text: nj

text: k

text: j

text: pj

text: k

text: together

text: with

text: the

text: requirement

text: sup

text: c

text: (m

text: (z))nk

text: k

text: pk

text: z

text: D

text: k

text: It

text: is

text: easy

text: to

text: construct

text: the

text: sequences

text: c

text: satisfying

text: the

text: above

text: conditions

text: at

text: k

text: k

text: each

text: stage.

text: Then

text: the

text: series

text: c

text: m

text: (z)

text: nj

text: j

text: pj

text: is

text: compactly

text: convergent

text: (recall

text: D

text: ’s

text: are

text: relatively

text: compact)

text: on

text: D

text: and

text: thus

text: deﬁnes

text: a

text: holomor-

text: j

text: phic

text: function

text: f

text: (z)

text: on

text: D.

text: As

text: f

text: (pn)

text: n

text: we

text: must

text: have

text: lim

text: sup

text: f

text: (z)

text: with

text: which

text: we

text: p

text: p

text: p

text: z

text: p

text: have

text: attained

text: our

text: goal

text: of

text: checking

text: out

text: that

text: domains

text: of

text: convergence

text: of

text: power

text: series

text: are

text: indeed

text: domains

text: of

text: holomorphy.


text: homeomorphic to some Euclidean space. A line-free closed convex set C may be expressed as a sum of its characteristic cone and the convex hull of its extreme points ext(C). We shall not spend any further space about such results or notions of convex bodies, which are strictly speak- ing not needed here. However, they may provide helpful background and in this regard we ﬁnd the availability of good expositions of basics of convex bodies in suﬃcient abundance, including the systematic encyclopaedic volume [10] which however for the most part restricts attention to compact convex bodies. Basics of unbounded convex bodies with proofs and references can be found in [6] or from the much older article [20]. Among basic examples of bounded convex sets are balls with respect to any norm. Of course all norms on the ﬁnite dimensional V are equivalent; but they are far from being aﬃnely equivalent – note that convexity is preserved by invertible aﬃne maps of V – in the sense that one cannot be obtained from the other by an aﬃne change of variables. Convexity may also be considered on spheres. To introduce this quickly, R − consider the unit sphere centered at the origin in N denoted SN 1 and a subset C thereof. If the cone consisting of rays through the origin and passing through points p, as p varies through R C, happens to be a convex cone in N, then we say that C is a spherically convex subset of − SN 1. A spherical-geometric way of checking convexity of a subset A of the sphere, is to check ∈ (cid:54) ± for every pair of points x, y A with y = x that, the set A contains the smaller arc of the − great circle on SN 1 connecting x and y. We next pass onto the notion of convex functions. → −∞ ∞ Deﬁnition 3.1. Let X be a convex set. A function f : X ( , + ] is termed convex if ≤ f(λ x + λ x ) λ f(x ) + λ f(x ) 1 1 2 2 1 1 2 2 ∈ for all pairs of positive numbers λ , λ with λ +λ = 1 and x , x X. Equivalently, a function 1 2 1 2 1 2 is convex iﬀ its epigraph { ∈ ⊕ R ∈ ≥ } (x, t) V : x X, t f(x) is a convex set. Theorem 3.2. If f is a convex function on V , then { ∈ ∞} X = x V : f(x) < is a convex set and f is continuous in the relative interior of X i.e, in the interior of X in ah(X). Remark 3.3. It is not always possible to redeﬁne f at boundary points of X in ah(X), so as to −∞ ∞ have f become continuous with values in ( , + ]. This problem is redressed by taking the lower-semicontinuous regularization. ∈ Proposition 3.4. Let f be a convex function on V . Deﬁne for all x V : f (x) = lim inf f(y) 1 → y x ≤ { ∈ Then f is convex and f (x) f(x) for all x, with equality if x lies in the interior of X = x 1 ∞} 1 \ V : f(x) < in ah(X) or interior in V X. The function f is lower semi-continuous and is 1 termed the lower semi-continuous regularization of f. If f is not given to be deﬁned on all of V but given on a convex set X, we ﬁrst extend by ∞ \ setting its values equal to + at all points where it is not apriori given i.e., on V X; the above proposition then applies to furnish its lower semicontinuous regularization. The role of lower semi-continuity here is explained as follows. While the epigraph of a function f is convex iﬀ its 35

text: homeomorphic

text: to

text: some

text: Euclidean

text: space.

text: A

text: line-free

text: closed

text: convex

text: set

text: C

text: may

text: be

text: expressed

text: as

text: a

text: sum

text: of

text: its

text: characteristic

text: cone

text: and

text: the

text: convex

text: hull

text: of

text: its

text: extreme

text: points

text: ext(C).

text: We

text: shall

text: not

text: spend

text: any

text: further

text: space

text: about

text: such

text: results

text: or

text: notions

text: of

text: convex

text: bodies,

text: which

text: are

text: strictly

text: speak-

text: ing

text: not

text: needed

text: here.

text: However,

text: they

text: may

text: provide

text: helpful

text: background

text: and

text: in

text: this

text: regard

text: we

text: ﬁnd

text: the

text: availability

text: of

text: good

text: expositions

text: of

text: basics

text: of

text: convex

text: bodies

text: in

text: suﬃcient

text: abundance,

text: including

text: the

text: systematic

text: encyclopaedic

text: volume

text: which

text: however

text: for

text: the

text: most

text: part

text: restricts

text: attention

text: to

text: compact

text: convex

text: bodies.

text: Basics

text: of

text: unbounded

text: convex

text: bodies

text: with

text: proofs

text: and

text: references

text: can

text: be

text: found

text: in

text: or

text: from

text: the

text: much

text: older

text: article

text: Among

text: basic

text: examples

text: of

text: bounded

text: convex

text: sets

text: are

text: balls

text: with

text: respect

text: to

text: any

text: norm.

text: Of

text: course

text: all

text: norms

text: on

text: the

text: ﬁnite

text: dimensional

text: V

text: are

text: equivalent;

text: but

text: they

text: are

text: far

text: from

text: being

text: aﬃnely

text: equivalent

text: note

text: that

text: convexity

text: is

text: preserved

text: by

text: invertible

text: aﬃne

text: maps

text: of

text: V

text: in

text: the

text: sense

text: that

text: one

text: cannot

text: be

text: obtained

text: from

text: the

text: other

text: by

text: an

text: aﬃne

text: change

text: of

text: variables.

text: Convexity

text: may

text: also

text: be

text: considered

text: on

text: spheres.

text: To

text: introduce

text: this

text: quickly,

text: R

text: consider

text: the

text: unit

text: sphere

text: centered

text: at

text: the

text: origin

text: in

text: N

text: denoted

text: SN

text: and

text: a

text: subset

text: C

text: thereof.

text: If

text: the

text: cone

text: consisting

text: of

text: rays

text: through

text: the

text: origin

text: and

text: passing

text: through

text: points

text: p,

text: as

text: p

text: varies

text: through

text: R

text: C,

text: happens

text: to

text: be

text: a

text: convex

text: cone

text: in

text: N,

text: then

text: we

text: say

text: that

text: C

text: is

text: a

text: spherically

text: convex

text: subset

text: of

text: SN

text: A

text: spherical-geometric

text: way

text: of

text: checking

text: convexity

text: of

text: a

text: subset

text: A

text: of

text: the

text: sphere,

text: is

text: to

text: check

text: for

text: every

text: pair

text: of

text: points

text: x,

text: y

text: A

text: with

text: y

text: x

text: that,

text: the

text: set

text: A

text: contains

text: the

text: smaller

text: arc

text: of

text: the

text: great

text: circle

text: on

text: SN

text: connecting

text: x

text: and

text: y.

text: We

text: next

text: pass

text: onto

text: the

text: notion

text: of

text: convex

text: functions.

text: Deﬁnition

text: Let

text: X

text: be

text: a

text: convex

text: set.

text: A

text: function

text: f

text: X

text: is

text: termed

text: convex

text: if

text: f(λ

text: x

text: λ

text: x

text: λ

text: f(x

text: λ

text: f(x

text: for

text: all

text: pairs

text: of

text: positive

text: numbers

text: λ

text: λ

text: with

text: λ

text: +λ

text: and

text: x

text: x

text: X.

text: Equivalently,

text: a

text: function

text: is

text: convex

text: iﬀ

text: its

text: epigraph

text: R

text: (x,

text: t)

text: V

text: x

text: X,

text: t

text: f(x)

text: is

text: a

text: convex

text: set.

text: Theorem

text: If

text: f

text: is

text: a

text: convex

text: function

text: on

text: V

text: then

text: X

text: x

text: V

text: f(x)

text: is

text: a

text: convex

text: set

text: and

text: f

text: is

text: continuous

text: in

text: the

text: relative

text: interior

text: of

text: X

text: i.e,

text: in

text: the

text: interior

text: of

text: X

text: in

text: ah(X).

text: Remark

text: It

text: is

text: not

text: always

text: possible

text: to

text: redeﬁne

text: f

text: at

text: boundary

text: points

text: of

text: X

text: in

text: ah(X),

text: so

text: as

text: to

text: have

text: f

text: become

text: continuous

text: with

text: values

text: in

text: This

text: problem

text: is

text: redressed

text: by

text: taking

text: the

text: lower-semicontinuous

text: regularization.

text: Proposition

text: Let

text: f

text: be

text: a

text: convex

text: function

text: on

text: V

text: Deﬁne

text: for

text: all

text: x

text: V

text: f

text: (x)

text: lim

text: inf

text: f(y)

text: y

text: x

text: Then

text: f

text: is

text: convex

text: and

text: f

text: (x)

text: f(x)

text: for

text: all

text: x,

text: with

text: equality

text: if

text: x

text: lies

text: in

text: the

text: interior

text: of

text: X

text: x

text: V

text: f(x)

text: in

text: ah(X)

text: or

text: interior

text: in

text: V

text: X.

text: The

text: function

text: f

text: is

text: lower

text: semi-continuous

text: and

text: is

text: termed

text: the

text: lower

text: semi-continuous

text: regularization

text: of

text: f.

text: If

text: f

text: is

text: not

text: given

text: to

text: be

text: deﬁned

text: on

text: all

text: of

text: V

text: but

text: given

text: on

text: a

text: convex

text: set

text: X,

text: we

text: ﬁrst

text: extend

text: by

text: setting

text: its

text: values

text: equal

text: to

text: at

text: all

text: points

text: where

text: it

text: is

text: not

text: apriori

text: given

text: i.e.,

text: on

text: V

text: X;

text: the

text: above

text: proposition

text: then

text: applies

text: to

text: furnish

text: its

text: lower

text: semicontinuous

text: regularization.

text: The

text: role

text: of

text: lower

text: semi-continuity

text: here

text: is

text: explained

text: as

text: follows.

text: While

text: the

text: epigraph

text: of

text: a

text: function

text: f

text: is

text: convex

text: iﬀ

text: its


text: epigraph is convex, the epigraph is closed iﬀ f is lower semi-continuous. This will be important in the subsection on the Legendre transform. ⊂ Deﬁnition 3.5. Let E V . The indicator function I is the function whose value at points of ∞ E E is set equal to 0 and equal to + at all points outside E. Such a function is convex precisely when E is convex. The following four results go by the name of Hahn – Banach theorems. (cid:54)∈ Theorem 3.6. Let D be a convex domain in V . If x D, there is an aﬃne hyperplane H ∈ ∩ ∅ 0 such that x H but H D = . Thus there is an aﬃne function f on V with f(x ) = 0 > f(x) ∈ 0 0 for all x D. (cid:54)∈ Corollary 3.7. Let X be a closed convex subset of V . If x X, there is an aﬃne hyperplane 0 ≤ containing x which does not intersect X i.e., there is an aﬃne function f with f(x) 0 < f(x ) ∈ 0 0 for all x X. Corollary 3.8. If X is a closed convex subset of V and if y is on the boundary of X, then one ≤ ∈ can ﬁnd a non-constant aﬃne function f such that f(x) 0 = f(y) for all x X. The aﬃne { ∈ } hyperplane x V : f(x) = 0 is called a supporting hyperplane of X. Corollary 3.9. An open (resp. closed) convex set K in a ﬁnite dimensional vector space is the intersection of the open (resp. closed) half-spaces containing it. As a closed convex set is the intersection of its supporting half-spaces, such a set can alternatively be described by specifying the position of its supporting hyperplanes, given their gradient vectors. This is captured by the support function introduced in deﬁnition (1.6). The geometric meaning of the support function is: for a unit vector u with h(u) ﬁnite, the number h(u) is the signed distance of the supporting hyperplane to C with normal vector u, from the origin; the distance is negative if and only if u points into the open half-space containing the origin. From the · (cid:104) ·(cid:105) deﬁnition, it is straight-forward to check that h ( ) = z, is a linear functional iﬀ C is a C ≥ singleton. More importantly, h is positively homogeneous: h(λu) = λh(u) for all λ 0 and is sub-additive: ≤ h(u + v) h(u) + h(v). These conditions constitute what is sometimes referred to as sub-linearity, from which it follows ∈ R \ in particular that h is a convex function. If x N C, a separation theorem yields the existence (cid:104) (cid:105) of a vector u with x, u > h(u ). The support function of a convex set C may also be deﬁned 0 0 0 as the Legendre transform of its indicator function I ; the Legendre transform being deﬁned in C the following subsection. R → R Deﬁnition 3.10. Let f : N any function. The Legendre transform (= Fenchel – Legendre transform), also called the convex conjugate, of f, is deﬁned by ∗ {(cid:104) (cid:105) − } f (y) = sup x, y f(x) . x 36

text: epigraph

text: is

text: convex,

text: the

text: epigraph

text: is

text: closed

text: iﬀ

text: f

text: is

text: lower

text: semi-continuous.

text: This

text: will

text: be

text: important

text: in

text: the

text: subsection

text: on

text: the

text: Legendre

text: transform.

text: Deﬁnition

text: Let

text: E

text: V

text: The

text: indicator

text: function

text: I

text: is

text: the

text: function

text: whose

text: value

text: at

text: points

text: of

text: E

text: E

text: is

text: set

text: equal

text: to

text: and

text: equal

text: to

text: at

text: all

text: points

text: outside

text: E.

text: Such

text: a

text: function

text: is

text: convex

text: precisely

text: when

text: E

text: is

text: convex.

text: The

text: following

text: four

text: results

text: go

text: by

text: the

text: name

text: of

text: Hahn

text: Banach

text: theorems.

text: Theorem

text: Let

text: D

text: be

text: a

text: convex

text: domain

text: in

text: V

text: If

text: x

text: D,

text: there

text: is

text: an

text: aﬃne

text: hyperplane

text: H

text: such

text: that

text: x

text: H

text: but

text: H

text: D

text: Thus

text: there

text: is

text: an

text: aﬃne

text: function

text: f

text: on

text: V

text: with

text: f(x

text: f(x)

text: for

text: all

text: x

text: D.

text: Corollary

text: Let

text: X

text: be

text: a

text: closed

text: convex

text: subset

text: of

text: V

text: If

text: x

text: X,

text: there

text: is

text: an

text: aﬃne

text: hyperplane

text: containing

text: x

text: which

text: does

text: not

text: intersect

text: X

text: i.e.,

text: there

text: is

text: an

text: aﬃne

text: function

text: f

text: with

text: f(x)

text: f(x

text: for

text: all

text: x

text: X.

text: Corollary

text: If

text: X

text: is

text: a

text: closed

text: convex

text: subset

text: of

text: V

text: and

text: if

text: y

text: is

text: on

text: the

text: boundary

text: of

text: X,

text: then

text: one

text: can

text: ﬁnd

text: a

text: non-constant

text: aﬃne

text: function

text: f

text: such

text: that

text: f(x)

text: f(y)

text: for

text: all

text: x

text: X.

text: The

text: aﬃne

text: hyperplane

text: x

text: V

text: f(x)

text: is

text: called

text: a

text: supporting

text: hyperplane

text: of

text: X.

text: Corollary

text: An

text: open

text: (resp.

text: closed)

text: convex

text: set

text: K

text: in

text: a

text: ﬁnite

text: dimensional

text: vector

text: space

text: is

text: the

text: intersection

text: of

text: the

text: open

text: (resp.

text: closed)

text: half-spaces

text: containing

text: it.

text: As

text: a

text: closed

text: convex

text: set

text: is

text: the

text: intersection

text: of

text: its

text: supporting

text: half-spaces,

text: such

text: a

text: set

text: can

text: alternatively

text: be

text: described

text: by

text: specifying

text: the

text: position

text: of

text: its

text: supporting

text: hyperplanes,

text: given

text: their

text: gradient

text: vectors.

text: This

text: is

text: captured

text: by

text: the

text: support

text: function

text: introduced

text: in

text: deﬁnition

text: The

text: geometric

text: meaning

text: of

text: the

text: support

text: function

text: is:

text: for

text: a

text: unit

text: vector

text: u

text: with

text: h(u)

text: ﬁnite,

text: the

text: number

text: h(u)

text: is

text: the

text: signed

text: distance

text: of

text: the

text: supporting

text: hyperplane

text: to

text: C

text: with

text: normal

text: vector

text: u,

text: from

text: the

text: origin;

text: the

text: distance

text: is

text: negative

text: if

text: and

text: only

text: if

text: u

text: points

text: into

text: the

text: open

text: half-space

text: containing

text: the

text: origin.

text: From

text: the

text: deﬁnition,

text: it

text: is

text: straight-forward

text: to

text: check

text: that

text: h

text: z,

text: is

text: a

text: linear

text: functional

text: iﬀ

text: C

text: is

text: a

text: C

text: singleton.

text: More

text: importantly,

text: h

text: is

text: positively

text: homogeneous:

text: h(λu)

text: λh(u)

text: for

text: all

text: λ

text: and

text: is

text: sub-additive:

text: h(u

text: v)

text: h(u)

text: h(v).

text: These

text: conditions

text: constitute

text: what

text: is

text: sometimes

text: referred

text: to

text: as

text: sub-linearity,

text: from

text: which

text: it

text: follows

text: R

text: in

text: particular

text: that

text: h

text: is

text: a

text: convex

text: function.

text: If

text: x

text: N

text: C,

text: a

text: separation

text: theorem

text: yields

text: the

text: existence

text: of

text: a

text: vector

text: u

text: with

text: x,

text: u

text: h(u

text: The

text: support

text: function

text: of

text: a

text: convex

text: set

text: C

text: may

text: also

text: be

text: deﬁned

text: as

text: the

text: Legendre

text: transform

text: of

text: its

text: indicator

text: function

text: I

text: the

text: Legendre

text: transform

text: being

text: deﬁned

text: in

text: C

text: the

text: following

text: subsection.

text: R

text: R

text: Deﬁnition

text: Let

text: f

text: N

text: any

text: function.

text: The

text: Legendre

text: transform

text: Fenchel

text: Legendre

text: transform),

text: also

text: called

text: the

text: convex

text: conjugate,

text: of

text: f,

text: is

text: deﬁned

text: by

text: f

text: (y)

text: sup

text: x,

text: y

text: f(x)

text: x


text: (cid:90) coeﬃcients of the power series deﬁning the function f, can be recovered from the knowledge of (cid:48) the values of f on the distinguished boundary of P by the formula: 1 − c = f(z)e i(k1θ1+...+kNθN)dθ . . . dθ K (2π)NrK 1 N [0,2π]N and consequently, we have the estimate | | | | ≤ 1 f T c K (2π)N rK (cid:88) Proof. Set zj = rjeiθj for each j = 1, . . . , N to write f(z) = f(r eiθ1, . . . , r eiθN ) = c rJei(j1θ1+...+jNθN) 1 N J (cid:90) and integrate with respect to each of the variables θ on [0, 2π] to get j − (cid:90) f(z)e i(k1θ1+...+kNθN)dθ . . . dθ (cid:88) 1 N [0,2π]N − − = c rJ ei(j1 k1)θ1+...+(jN kN)θN dθ . . . dθ . J 1 N [0,2π]N where the interchange of integral and summation on the right is justiﬁed by the uniform con- (cid:48) vergence of our power series on the boundary of P . The integral appearing on the right in the last equation is zero except when J = K in which case it is (2π)N. The formulae in assertion (cid:3) now follow. C∗ Deﬁnition 2.8. Let z0 be any point of ( )N. The (open) polydisc centered at the origin with | | | | | | polyradius ( z0 , z0 , . . . , z0 ) is called the po(cid:80)lydisc spanned by the point z0. 1 2 N We may rephrase Abel’s lemma as follows. Let P be a polydisc and w a point of the distinguished boundary of P . If the power series f(z) = c zJ converges (unconditionally) at w, then it J converges compactly on P . Stated diﬀerently, if f converges at a point w, then it converges compactly on the polydisc spanned by w. This means that the interior of the set of convergence of the general power series f which we denoted B0, can be expressed as the union of the (concentric) polydiscs spanned by points of B and subsequently that B0 must be connected. This ﬁnishes the pending veriﬁcation that B0 is indeed a domain. In fact, we may note more here: B0 is what is known as a Reinhardt domain, indeed a ‘complete Reinhardt domain’ as deﬁned below and in particular therefore, a contractible domain. C ∈ Deﬁnition 2.9. A domain D in N is termed Reinhardt (about the origin) if z D entails ∈ ∈ R that (eiθ1z1, . . . , eiθN zN) D for all possiblCe choices of (θ1, . . . , θN) N∈. Such a domain is also said to be multi-circular. A domain D in N is said to be circular if z D entails (only) that ∈ ∈ R (eiθz , . . . , eiθz ) D for all θ ; it is said to be complete circular if it admits an action by 1 N∈ ∈ ∈ the disc i.e., z D entails that (λz , . . . , λz ) D for all λ ∆; complete circular domains are 1 N sometimes also referred to as complex star-like domains and we note in passing that being star- like with respect to the origin, all complete circular domains are contractible domains. Likewise a Reinhardt domain is said to be complete if it is invariant under the action of the closed unit ∈ ∈ poydisc by coordinate-wise multiplication i.e., z D entails that (λ z , . . . , λ z ) D for all 1 1 N N ∈ N choices of (λ , . . . , λ ) ∆ . 1 N 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: coeﬃcients

text: of

text: the

text: power

text: series

text: deﬁning

text: the

text: function

text: f,

text: can

text: be

text: recovered

text: from

text: the

text: knowledge

text: of

text: the

text: values

text: of

text: f

text: on

text: the

text: distinguished

text: boundary

text: of

text: P

text: by

text: the

text: formula:

text: c

text: f(z)e

text: dθ

text: K

text: N

text: and

text: consequently,

text: we

text: have

text: the

text: estimate

text: f

text: T

text: c

text: K

text: rK

text: Proof.

text: Set

text: zj

text: rjeiθj

text: for

text: each

text: j

text: N

text: to

text: write

text: f(z)

text: f(r

text: r

text: eiθN

text: c

text: N

text: J

text: and

text: integrate

text: with

text: respect

text: to

text: each

text: of

text: the

text: variables

text: θ

text: on

text: to

text: get

text: j

text: f(z)e

text: dθ

text: N

text: c

text: rJ

text: kN)θN

text: dθ

text: dθ

text: J

text: N

text: where

text: the

text: interchange

text: of

text: integral

text: and

text: summation

text: on

text: the

text: right

text: is

text: justiﬁed

text: by

text: the

text: uniform

text: con-

text: vergence

text: of

text: our

text: power

text: series

text: on

text: the

text: boundary

text: of

text: P

text: The

text: integral

text: appearing

text: on

text: the

text: right

text: in

text: the

text: last

text: equation

text: is

text: zero

text: except

text: when

text: J

text: K

text: in

text: which

text: case

text: it

text: is

text: The

text: formulae

text: in

text: assertion

text: now

text: follow.

text: C∗

text: Deﬁnition

text: Let

text: be

text: any

text: point

text: of

text: )N.

text: The

text: (open)

text: polydisc

text: centered

text: at

text: the

text: origin

text: with

text: polyradius

text: is

text: called

text: the

text: spanned

text: by

text: the

text: point

text: N

text: We

text: may

text: rephrase

text: Abel’s

text: lemma

text: as

text: follows.

text: Let

text: P

text: be

text: a

text: polydisc

text: and

text: w

text: a

text: point

text: of

text: the

text: distinguished

text: boundary

text: of

text: P

text: If

text: the

text: power

text: series

text: f(z)

text: c

text: zJ

text: converges

text: (unconditionally)

text: at

text: w,

text: then

text: it

text: J

text: converges

text: compactly

text: on

text: P

text: Stated

text: diﬀerently,

text: if

text: f

text: converges

text: at

text: a

text: point

text: w,

text: then

text: it

text: converges

text: compactly

text: on

text: the

text: polydisc

text: spanned

text: by

text: w.

text: This

text: means

text: that

text: the

text: interior

text: of

text: the

text: set

text: of

text: convergence

text: of

text: the

text: general

text: power

text: series

text: f

text: which

text: we

text: denoted

text: can

text: be

text: expressed

text: as

text: the

text: union

text: of

text: the

text: (concentric)

text: polydiscs

text: spanned

text: by

text: points

text: of

text: B

text: and

text: subsequently

text: that

text: must

text: be

text: connected.

text: This

text: ﬁnishes

text: the

text: pending

text: veriﬁcation

text: that

text: is

text: indeed

text: a

text: domain.

text: In

text: fact,

text: we

text: may

text: note

text: more

text: here:

text: is

text: what

text: is

text: known

text: as

text: a

text: Reinhardt

text: domain,

text: indeed

text: a

text: ‘complete

text: Reinhardt

text: domain’

text: as

text: deﬁned

text: below

text: and

text: in

text: particular

text: therefore,

text: a

text: contractible

text: domain.

text: C

text: Deﬁnition

text: A

text: domain

text: D

text: in

text: N

text: is

text: termed

text: Reinhardt

text: (about

text: the

text: origin)

text: if

text: z

text: D

text: entails

text: R

text: that

text: eiθN

text: zN)

text: D

text: for

text: all

text: possiblCe

text: choices

text: of

text: θN)

text: N∈.

text: Such

text: a

text: domain

text: is

text: also

text: said

text: to

text: be

text: multi-circular.

text: A

text: domain

text: D

text: in

text: N

text: is

text: said

text: to

text: be

text: circular

text: if

text: z

text: D

text: entails

text: (only)

text: that

text: R

text: (eiθz

text: eiθz

text: D

text: for

text: all

text: θ

text: it

text: is

text: said

text: to

text: be

text: complete

text: circular

text: if

text: it

text: admits

text: an

text: action

text: by

text: N∈

text: the

text: disc

text: i.e.,

text: z

text: D

text: entails

text: that

text: (λz

text: λz

text: D

text: for

text: all

text: λ

text: complete

text: circular

text: domains

text: are

text: N

text: sometimes

text: also

text: referred

text: to

text: as

text: complex

text: star-like

text: domains

text: and

text: we

text: note

text: in

text: passing

text: that

text: being

text: star-

text: like

text: with

text: respect

text: to

text: the

text: origin,

text: all

text: complete

text: circular

text: domains

text: are

text: contractible

text: domains.

text: Likewise

text: a

text: Reinhardt

text: domain

text: is

text: said

text: to

text: be

text: complete

text: if

text: it

text: is

text: invariant

text: under

text: the

text: action

text: of

text: the

text: closed

text: unit

text: poydisc

text: by

text: coordinate-wise

text: multiplication

text: i.e.,

text: z

text: D

text: entails

text: that

text: (λ

text: z

text: λ

text: z

text: D

text: for

text: all

text: N

text: N

text: N

text: choices

text: of

text: (λ

text: λ

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 * et al. / Nuclear Physics A 00 (2020) 1–8 T, which is compatible with the Debye screening expectation for heavy quark interaction. The question is what is the relation between the lattice calculated free energy and the heavy quark potential V(T, r) which controls the quarkonium dissociation. By solving the Schroedinger equation for the heavy quark pair, the (cid:39) dissociation temperature for J/ψ in the limit of V = F is TJ/ψ 1.2Tc with Tc being the critical temperature of the deconﬁnement phase transition, while in the other limit of V = U J/ψs can survive in the deconﬁned (cid:39) plasma up to the temperature TJ/ψ 2.1Tc, showing a much more strong binding. Phenomenologically, the comparison between a model calculation and the experimental data on excited Υ states is sensitive to the choice of the heavy quark potential [21]. On the other hand, based on the directly calculated quarkonium correlator G(r, τ) from lattice QCD, the in-medium quarkonium spectrum can be evaluated by the Maximum Entropy Method (with still a large uncertainty in the reconstruction), and the dissociation temperature can be estimated from the disappearance of the peak of the spectral function. For J/ψ, the melting point is located (cid:39) at TJ/ψ 1.5Tc from the recent hot QCD simulation [22]. One should note that the above potential analysis assumes the medium to be stable and isotropic. How- ever, in a realistic heavy ion collision, due to the fast expansion and non-zero viscosity of the system, the produced medium would exhibit a local anisotropy in momentum space. Stronger attraction is found to be aligned along the longitudinal direction in an anisotropic medium [23], which will enhance the quarkonium surviving temperature compared to the isotropic case. The relativistic correction to the potential model will − also enhance the J/ψ dissociation temperature by 7% 13% depending on the choice of the potential [24]. Apart from the screening of the attractive binding between heavy quarks, the bound states would also suﬀer a scattering dissociation by the medium constituents, which is not captured by the above analysis and should manifest as a thermal width in the spectral function. From thermal ﬁeld theory, a static heavy quark potential at ﬁnite temperature is recently evaluated by Fourier transformation of the hard thermal loop resumed gluon propagator [25]. Due to the complexit(cid:29)y(cid:90)of the∼gluon self energy, an imaginary part is automatically developed for the heavy quark potential at T 1/r mD, ∞ VHTL(r ∼ m1 ) = −43 αrs (e−mDr + mD) + i83αsT dt(1 − sinm(mrDtrt))(t2 +t 1)2 , (6) D 0 D where mD is the gluon Debye screening mass(cid:90). This imaginary part smears the spectral peaks and leads to a ﬁnite thermal width reﬂecting the thermal medium eﬀects, † Γ = Ψ [ImV]Ψ, (7) with Ψ being the in-medium charmonium wave function. The underlying physics of this ﬁnite width is interpreted as the Landau damping of the low frequency gauge ﬁeld. In terms of the nonrelativistic ﬁeld theory pNRQCD, the heavy quark potential is evaluated, and an additional color singlet-to-octet thermal break-up mechanism is found to contribute to the quarkonium thermal width [26], especially in the temper- ature range where m is smaller than the binding energy. Motivated by the eﬀective ﬁeld theory calculation, D a lattice regularized QCD simulation based on NRQCD Lagrangian is recently performed [27]. The heavy quark potential is extracted from the simulated spectral function, where the spectral peak and width are respectively related to the real and imaginary part of the potential. It shows that, the imaginary part of the potential ImV is compatible with the perturbative result [25], and the real part ReV from HotQCD collabo- ration with N = 2+1 ﬂavors is closer to the free energy F. By parameterizing the potential in a generalized f Gauss law approach and putting it into the Schroedinger equation, the dissociation temperature for J/ψ is (cid:39) TJ/ψ 1.37Tc [28]. To understand the lattice QCD result from a point of view of many-body physics, a non-perturbative T-matrix approach is developed [29]. In this approach, by ﬁtting the lattice calculated free energy, the heavy quark potential is recently evaluated [30] by taking a diagrammatic treatment through deﬁning the potential within a scattering equation. The extracted potential is found to be in between the free energy F and the internal energy U but very close to U. Interestingly, the derivative of the complex potential, namely the binding force between the two heavy quarks, is stronger than both from free energy and internal energy, implying a remnant conﬁning string tension in the medium related with a long-range correlation. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: et

text: al.

font: VCKFCA+rtxi

text: Nuclear

text: Physics

text: A

text: T,

text: which

text: is

text: compatible

text: with

text: the

text: Debye

text: screening

text: expectation

text: for

text: heavy

text: quark

text: interaction.

text: The

text: question

text: is

text: what

text: is

text: the

text: relation

text: between

text: the

text: lattice

text: calculated

text: free

text: energy

text: and

text: the

text: heavy

text: quark

text: potential

text: V(T,

text: r)

text: which

text: controls

text: the

text: quarkonium

text: dissociation.

text: By

text: solving

text: the

text: Schroedinger

text: equation

text: for

text: the

text: heavy

text: quark

text: pair,

text: the

font: SCZOPQ+txsy

text: dissociation

text: temperature

text: for

text: J/ψ

font: HWIXRK+rtxmi

text: in

text: the

text: limit

text: of

text: V

font: FHHMAQ+rtxr

text: F

text: is

text: TJ/ψ

text: with

text: Tc

text: being

text: the

text: critical

text: temperature

text: of

text: the

text: deconﬁnement

text: phase

text: transition,

text: while

text: in

text: the

text: other

text: limit

text: of

text: V

text: U

text: J/ψs

text: can

text: survive

text: in

text: the

text: deconﬁned

text: plasma

text: up

text: to

text: the

text: temperature

text: TJ/ψ

text: showing

text: a

text: much

text: more

text: strong

text: binding.

text: Phenomenologically,

text: the

text: comparison

text: between

text: a

text: model

text: calculation

text: and

text: the

text: experimental

text: data

text: on

text: excited

text: Υ

text: states

text: is

text: sensitive

text: to

text: the

text: choice

text: of

text: the

text: heavy

text: quark

text: potential

text: On

text: the

text: other

text: hand,

text: based

text: on

text: the

text: directly

text: calculated

text: quarkonium

text: correlator

text: G(r,

text: τ)

text: from

text: lattice

text: QCD,

text: the

text: in-medium

text: quarkonium

text: spectrum

text: can

text: be

text: evaluated

text: by

text: the

text: Maximum

text: Entropy

text: Method

text: (with

text: still

text: a

text: large

text: uncertainty

text: in

text: the

text: reconstruction),

text: and

text: the

text: dissociation

text: temperature

text: can

text: be

text: estimated

text: from

text: the

text: disappearance

text: of

text: the

text: peak

text: of

text: the

text: spectral

text: function.

text: For

text: J/ψ,

text: the

text: melting

text: point

text: is

text: located

text: at

text: TJ/ψ

text: from

text: the

text: recent

text: hot

text: QCD

text: simulation

text: One

text: should

text: note

text: that

text: the

text: above

text: potential

text: analysis

text: assumes

text: the

text: medium

text: to

text: be

text: stable

text: and

text: isotropic.

text: How-

text: ever,

text: in

text: a

text: realistic

text: heavy

text: ion

text: collision,

text: due

text: to

text: the

text: fast

text: expansion

text: and

text: non-zero

text: viscosity

text: of

text: the

text: system,

text: the

text: produced

text: medium

text: would

text: exhibit

text: a

text: local

text: anisotropy

text: in

text: momentum

text: space.

text: Stronger

text: attraction

text: is

text: found

text: to

text: be

text: aligned

text: along

text: the

text: longitudinal

text: direction

text: in

text: an

text: anisotropic

text: medium

text: which

text: will

text: enhance

text: the

text: quarkonium

text: surviving

text: temperature

text: compared

text: to

text: the

text: isotropic

text: case.

text: The

text: relativistic

text: correction

text: to

text: the

text: potential

text: model

text: will

text: also

text: enhance

text: the

text: J/ψ

text: dissociation

text: temperature

text: by

text: depending

text: on

text: the

text: choice

text: of

text: the

text: potential

text: Apart

text: from

text: the

text: screening

text: of

text: the

text: attractive

text: binding

text: between

text: heavy

text: quarks,

text: the

text: bound

text: states

text: would

text: also

text: suﬀer

text: a

text: scattering

text: dissociation

text: by

text: the

text: medium

text: constituents,

text: which

text: is

text: not

text: captured

text: by

text: the

text: above

text: analysis

text: and

text: should

text: manifest

text: as

text: a

text: thermal

text: width

text: in

text: the

text: spectral

text: function.

text: From

text: thermal

text: ﬁeld

text: theory,

text: a

text: static

text: heavy

text: quark

text: potential

text: at

text: ﬁnite

text: temperature

text: is

text: recently

text: evaluated

text: by

text: Fourier

text: transformation

text: of

text: the

text: hard

text: thermal

text: loop

text: resumed

text: gluon

text: propagator

text: Due

text: to

text: the

text: the∼gluon

text: self

text: energy,

text: an

text: imaginary

text: part

text: is

text: automatically

text: developed

text: for

text: the

text: heavy

text: quark

text: potential

text: at

text: T

text: mD,

text: VHTL(r

text: αrs

text: (e−mDr

text: mD)

text: +t

text: D

text: D

text: where

text: mD

text: is

text: the

text: gluon

text: Debye

text: screening

text: This

text: imaginary

text: part

text: smears

text: the

text: spectral

text: peaks

text: and

text: leads

text: to

text: a

text: ﬁnite

text: thermal

text: width

text: reﬂecting

text: the

text: thermal

text: medium

text: eﬀects,

text: Γ

text: Ψ

text: [ImV]Ψ,

text: with

text: Ψ

text: being

text: the

text: in-medium

text: charmonium

text: wave

text: function.

text: The

text: underlying

text: physics

text: of

text: this

text: ﬁnite

text: width

text: is

text: interpreted

text: as

text: the

text: Landau

text: damping

text: of

text: the

text: low

text: frequency

text: gauge

text: ﬁeld.

text: In

text: terms

text: of

text: the

text: nonrelativistic

text: ﬁeld

text: theory

text: pNRQCD,

text: the

text: heavy

text: quark

text: potential

text: is

text: evaluated,

text: and

text: an

text: additional

text: color

text: singlet-to-octet

text: thermal

text: break-up

text: mechanism

text: is

text: found

text: to

text: contribute

text: to

text: the

text: quarkonium

text: thermal

text: width

text: especially

text: in

text: the

text: temper-

text: ature

text: range

text: where

text: m

text: is

text: smaller

text: than

text: the

text: binding

text: energy.

text: Motivated

text: by

text: the

text: eﬀective

text: ﬁeld

text: theory

text: calculation,

text: D

text: a

text: lattice

text: regularized

text: QCD

text: simulation

text: based

text: on

text: NRQCD

text: Lagrangian

text: is

text: recently

text: performed

text: The

text: heavy

text: quark

text: potential

text: is

text: extracted

text: from

text: the

text: simulated

text: spectral

text: function,

text: where

text: the

text: spectral

text: peak

text: and

text: width

text: are

text: respectively

text: related

text: to

text: the

text: real

text: and

text: imaginary

text: part

text: of

text: the

text: potential.

text: It

text: shows

text: that,

text: the

text: imaginary

text: part

text: of

text: the

text: potential

text: ImV

text: is

text: compatible

text: with

text: the

text: perturbative

text: result

text: and

text: the

text: real

text: part

text: ReV

text: from

text: HotQCD

text: collabo-

text: ration

text: with

text: N

text: ﬂavors

text: is

text: closer

text: to

text: the

text: free

text: energy

text: F.

text: By

text: parameterizing

text: the

text: potential

text: in

text: a

text: generalized

text: f

text: Gauss

text: law

text: approach

text: and

text: putting

text: it

text: into

text: the

text: Schroedinger

text: equation,

text: the

text: dissociation

text: temperature

text: for

text: J/ψ

text: is

text: TJ/ψ

text: To

text: understand

text: the

text: lattice

text: QCD

text: result

text: from

text: a

text: point

text: of

text: view

text: of

text: many-body

text: physics,

text: a

text: non-perturbative

text: T-matrix

text: approach

text: is

text: developed

text: In

text: this

text: approach,

text: by

text: ﬁtting

text: the

text: lattice

text: calculated

text: free

text: energy,

text: the

text: heavy

text: quark

text: potential

text: is

text: recently

text: evaluated

text: by

text: taking

text: a

text: diagrammatic

text: treatment

text: through

text: deﬁning

text: the

text: potential

text: within

text: a

text: scattering

text: equation.

text: The

text: extracted

text: potential

text: is

text: found

text: to

text: be

text: in

text: between

text: the

text: free

text: energy

text: F

text: and

text: the

text: internal

text: energy

text: U

text: but

text: very

text: close

text: to

text: U.

text: Interestingly,

text: the

text: derivative

text: of

text: the

text: complex

text: potential,

text: namely

text: the

text: binding

text: force

text: between

text: the

text: two

text: heavy

text: quarks,

text: is

text: stronger

text: than

text: both

text: from

text: free

text: energy

text: and

text: internal

text: energy,

text: implying

text: a

text: remnant

text: conﬁning

text: string

text: tension

text: in

text: the

text: medium

text: related

text: with

text: a

text: long-range

text: correlation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 FIG. 9: (Color online) A squeezing transformation converts (a) a parallelogram to (b) a rectangle while it squeezes the entire proﬁle of Wigner function. ∗ α e2iφ sinh r. Note that any values of r and φ for the squeezing transformation S[α] can be used in Eqs. (41) and (42), as squeezing does not create non-Gaussianity. | (cid:105)(cid:104) | As an illustration, we show the case of the state f 0 0 + − | (cid:105)(cid:104) | (1 f) 2 2 in Figs. 8(c) and 8(d) using the above tests. B. Testing noisy non-Gaussian states We here investigate our test of genuine non- Gaussianity for noisy non-Gaussian states. In particu- lar, when a single-mode state undergoes a lossy Gaus- sian channel, the corresponding Wigner function becomes positive deﬁnite if its transmission rate η is below 50%. It is then interesting to know whether our non-Gaussianity (cid:80) test can detect noisy non-Gaussian states for η < 0.5. For this purpose, we consider a class of superposition | (cid:105) | (cid:105) states Ψ = N C n (N: truncation number) [26] N n=0 n as an input to a lossy channel. We then test the genuine non-Gaussianity for the output decohered states using − the Wigner function (s = 0) or the Q function (s = 1) (cid:80) in Eqs. (41) and (42). N In Fig. 10 (a), we plot (parallelogram test) for s=0 | (cid:105) the superposition states N C n with varying trun- n=0 n cation number N = 1 (blue solid), N = 2 (red dashed), N (cid:80) and N = 3 (black dotted). For each η, we optimize FIG. 10: (Color online) (a) (parallelogram test), (b) N N(cid:48) Ns=0 the coeﬃcientNs Cn to show a maximal 0 in the ﬁgures. s=0 (triangle test), and (c) s=−1 (parallelogram test), |re(cid:105)- We see that 0 is above the Gaussian bound 398/8 even spectively, optimized over a superposition state Nn=0 Cn n below η = 0.5 (magniﬁed view in the inset), demonstrat- for each η (transmission rate) under a loss channel. The ing a successful detection of non-Gaussianity for posi- curves from top to bottom represent the case of truncation number N = 3 (black dotted), N = 2 (red dashed) and N = 1 tive Wigner functions. The plots indicate that there N ≈ N(cid:48) (blue solid). > 8 2.324 and > 2, respectively, exist some superposition states for each N whose non- s 39/8 s represent genuine non-Gaussianity of the noisy non-Gaussian Gaussianity can be detected under transmission below N states. 50%. We have numerically found that > 8 appears 0 39/8 at a very low η. If we take into account the violation of ∼ size 0.001, the critical η turns out to be ηc = 0.590 powerful than the Wigner-function test (s = 0). How- (N = 1), ηc = 0.238 (N = 2), and ηc = 0.221 (N = 3), ever, they can also manifest genuine non-Gaussianity be- respectively. low η = 0.5 under the paralleogram test as shown in N(cid:48) On the other hand, under the triangle test , the de- Fig. 10 (c), with critical values η = 0.476 (N = 2) and 0 c tection becomes less eﬀective as shown in Fig. 10 (b). In η = 0.442 (N = 3) at the violation of size 0.001 (no c particular, the critical η does not go below 0.5 in con- violation for N = 1). We do not show the case of trian- N N(cid:48) trast to the rectangle test in Fig. 10 (a). Moving on gle test − , which does not detect non-Gaussianity. 0− s= 1 to the Q-function tests (s = 1), they are naturally less Thus, for the class of noisy non-Gaussian states consid- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: FIG.

text: (Color

label: caption

text: online)

text: A

text: squeezing

text: transformation

text: converts

text: (a)

text: a

text: parallelogram

text: to

text: (b)

text: a

text: rectangle

text: while

text: it

text: squeezes

text: the

text: entire

text: proﬁle

text: of

text: Wigner

text: function.

text: α

text: sinh

text: r.

text: Note

text: that

text: any

text: values

text: of

text: r

text: and

text: φ

text: for

text: the

text: squeezing

text: transformation

text: S[α]

text: can

text: be

text: used

text: in

text: Eqs.

text: and

text: as

text: squeezing

text: does

text: not

text: create

text: non-Gaussianity.

text: As

text: an

text: illustration,

text: we

text: show

text: the

text: case

text: of

text: the

text: state

text: f

text: f)

text: in

text: Figs.

text: and

text: using

text: the

text: above

text: tests.

text: B.

label: section

text: Testing

text: noisy

text: non-Gaussian

text: states

text: We

text: here

text: investigate

text: our

text: test

text: of

text: genuine

text: non-

text: Gaussianity

text: for

text: noisy

text: non-Gaussian

text: states.

text: In

text: particu-

text: lar,

text: when

text: a

text: single-mode

text: state

text: undergoes

text: a

text: lossy

text: Gaus-

text: sian

text: channel,

text: the

text: corresponding

text: Wigner

text: function

text: becomes

text: positive

text: deﬁnite

text: if

text: its

text: transmission

text: rate

text: η

text: is

text: below

text: It

text: is

text: then

text: interesting

text: to

text: know

text: whether

text: our

text: non-Gaussianity

text: test

text: can

text: detect

text: noisy

text: non-Gaussian

text: states

text: for

text: η

text: For

text: this

text: purpose,

text: we

text: consider

text: a

text: class

text: of

text: superposition

text: states

text: Ψ

text: N

text: C

text: n

text: (N:

text: truncation

text: number)

text: N

text: n

text: as

text: an

text: input

text: to

text: a

text: lossy

text: channel.

text: We

text: then

text: test

text: the

text: genuine

text: non-Gaussianity

text: for

text: the

text: output

text: decohered

text: states

text: using

text: the

text: Wigner

text: function

text: (s

text: or

text: the

text: Q

text: function

text: (s

text: in

text: Eqs.

text: and

text: N

text: In

text: Fig.

text: (a),

text: we

text: plot

text: (parallelogram

text: test)

text: for

text: the

text: superposition

text: states

text: N

text: C

text: n

text: with

text: varying

text: trun-

text: n

text: cation

text: number

text: N

text: (blue

text: solid),

text: N

text: (red

text: dashed),

text: N

text: and

text: N

text: (black

text: dotted).

text: For

text: each

text: η,

text: we

text: optimize

text: FIG.

text: (Color

text: online)

text: (a)

text: (parallelogram

text: test),

text: (b)

text: N

text: the

text: coeﬃcientNs

text: Cn

text: to

text: show

text: a

text: maximal

text: in

text: the

text: ﬁgures.

text: (triangle

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