text: ∗ This wavelet ψ(x) is based on a triplet of reals (a, b, x ) solving the system of equations (cid:0) (cid:1) − ∗ − ∗ ∗ x e (x )2/2 + ax + b = 0 (15a) ∗ − − ∗ (x )√2 1 e (x√)2/2 + a = 0 (15b) − − 3e 3/2 3a + b = 0, (15c) √ ∗ (cid:54) − ∗ ≈ ≈ where x = 3. A numerical solution exists with x 0.71, a 0.38, ≈ ∗ and b 0.28. Note that x is a transcendental (non-algebraic) number, as we will show. Now let − − ψ(x) = xe x2/2 + ax + b. √ ∗ − ψ(x) has a non-regular zero at x = x and a regular zero at x = 3. Its (cid:48) √ ± ∗ (cid:48)(cid:48) derivative ψ (x) has zeros at x = x Its second derivative ψ (x) has regular ± zeros at x = 0 and x = 3 and no non-regular zeros (see Figure 1). All ≥ derivatives of order n 3 have (algebraic) zeros given by the roots of the Hermite polynomial H (x). n+1 The wavelet ψ satisﬁes the weakened genericity condition. Indeed, the (cid:48) (cid:48)(cid:48) zeros of ψ are clearly not contained in those of ψ or ψ . Further, the zeroes of ψ(n) for n > 2 are zeroes of Hermite polynomials, which are algebraic, so ∗ (cid:48) ± ∗ they cannot contain x . Similarly, ψ has zeroes at x , which are not both ≥ contained in the zeroes of ψ, nor in the (algebraic) zeroes of ψ(n) for n 2. ≥ The zeroes of ψ(n), n 2, are algebraic and so cannot be contained in those (cid:48) ≥ of ψ or ψ , and are not contained in the zeroes of any other ψ(m) (m 2), since the zeroes of diﬀerent Hermites are never contained in each other (see Section ). √ Note however that ψ does not satisfy the original genericity condition, − (cid:48)(cid:48) since its only regular zero (at x = 3) is contained in the zero set of ψ . √ For c > 0 suﬃciently small and σ > 0 suﬃciently large, the functions ψ(w) − (cid:48)(cid:48) − and ψ(w) + cσ 2ψ (w) both have zeros only at x = 3. Thus the initial distributions δ(0) and δ(0) + cδ(2) cannot be distinguished by their zeros when the scaling σ is suﬃciently large. This shows that the weakened genericity condition is insuﬃcient for Marr’s conjecture to hold. ∗ It remains to show that x is transcendental (i.e., not an algebraic num- 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: wavelet

text: ψ(x)

text: is

text: based

text: on

text: a

text: triplet

text: of

text: reals

text: (a,

text: b,

text: x

text: solving

text: the

text: system

text: of

text: equations

text: x

text: e

text: (x

text: ax

text: b

text: (x

text: e

text: a

text: b

text: where

text: x

text: A

text: numerical

text: solution

text: exists

text: with

text: x

text: a

text: and

text: b

text: Note

text: that

text: x

text: is

text: a

text: transcendental

text: (non-algebraic)

text: number,

text: as

text: we

text: will

text: show.

text: Now

text: let

text: ψ(x)

text: xe

text: ax

text: b.

text: ψ(x)

text: has

text: a

text: non-regular

text: zero

text: at

text: x

text: x

text: and

text: a

text: regular

text: zero

text: at

text: x

text: Its

text: derivative

text: ψ

text: (x)

text: has

text: zeros

text: at

text: x

text: x

text: Its

text: second

text: derivative

text: ψ

text: (x)

text: has

text: regular

text: zeros

text: at

text: x

text: and

text: x

text: and

text: no

text: non-regular

text: zeros

text: (see

text: Figure

text: All

text: derivatives

text: of

text: order

text: n

text: have

text: (algebraic)

text: zeros

text: given

text: by

text: the

text: roots

text: of

text: the

text: Hermite

text: polynomial

text: H

text: (x).

text: The

text: wavelet

text: ψ

text: satisﬁes

text: the

text: weakened

text: genericity

text: condition.

text: Indeed,

text: the

text: zeros

text: of

text: ψ

text: are

text: clearly

text: not

text: contained

text: in

text: those

text: of

text: ψ

text: or

text: ψ

text: Further,

text: the

text: zeroes

text: of

text: ψ(n)

text: for

text: n

text: are

text: zeroes

text: of

text: Hermite

text: polynomials,

text: which

text: are

text: algebraic,

text: so

text: they

text: cannot

text: contain

text: x

text: Similarly,

text: ψ

text: has

text: zeroes

text: at

text: x

text: which

text: are

text: not

text: both

text: contained

text: in

text: the

text: zeroes

text: of

text: ψ,

text: nor

text: in

text: the

text: (algebraic)

text: zeroes

text: of

text: ψ(n)

text: for

text: n

text: The

text: zeroes

text: of

text: ψ(n),

text: n

text: are

text: algebraic

text: and

text: so

text: cannot

text: be

text: contained

text: in

text: those

text: of

text: ψ

text: or

text: ψ

text: and

text: are

text: not

text: contained

text: in

text: the

text: zeroes

text: of

text: any

text: other

text: ψ(m)

text: (m

text: since

text: the

text: zeroes

text: of

text: diﬀerent

text: Hermites

text: are

text: never

text: contained

text: in

text: each

text: other

text: (see

text: Section

text: Note

text: however

text: that

text: ψ

text: does

text: not

text: satisfy

text: the

text: original

text: genericity

text: condition,

text: since

text: its

text: only

text: regular

text: zero

text: (at

text: x

text: is

text: contained

text: in

text: the

text: zero

text: set

text: of

text: ψ

text: For

text: c

text: suﬃciently

text: small

text: and

text: σ

text: suﬃciently

text: large,

text: the

text: functions

text: ψ(w)

text: and

text: ψ(w)

text: cσ

text: 2ψ

text: (w)

text: both

text: have

text: zeros

text: only

text: at

text: x

text: Thus

text: the

text: initial

text: distributions

text: and

text: cannot

text: be

text: distinguished

text: by

text: their

text: zeros

text: when

text: the

text: scaling

text: σ

text: is

text: suﬃciently

text: large.

text: This

text: shows

text: that

text: the

text: weakened

text: genericity

text: condition

text: is

text: insuﬃcient

text: for

text: Marr’s

text: conjecture

text: to

text: hold.

text: It

text: remains

text: to

text: show

text: that

text: x

text: is

text: transcendental

text: (i.e.,

text: not

text: an

text: algebraic

text: num-

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text: 1.0 0.5 Ψ (cid:45)3 (cid:45)2 (cid:45)1 1 2 3 Ψ(cid:162)(cid:162) (cid:45) 0.5 (cid:45) 1.0 (cid:48)(cid:48) √ Figure 1: Graphs of ψ(x) and ψ (x). Note ψ(x) has a non-regular zero at ∗ √ − (cid:48)(cid:48) x = x and a regular zero at x = 3, while ψ (x) has regular zeros at ± x = 0 and x = 3 and no non-regular zeros (cid:0) (cid:1) ber). To this end, solving (15b) for a we have − ∗ − ∗ a = 1 (x )2 e( x )2/2. Now substituting for a in (15a) and solving for b: ∗ − ∗ b = (x )3e( x )2/2. (cid:0) (cid:1) Now substituting in (15c) and rearranging, √ √ − ∗ − ∗ − ∗ 3e 3/2+(x )2/2 = 3 1 (x )2 (x )3. ∗ If x were algebraic then the left side would be transcendental (since the expo- nentials of non-zero algebraics are transcendental by the Hermite-Lindemann theorem) while the right side would be algebraic, which would give a contra- ∗ diction. Therefore x is transcendental. 3.2 Ricker wavelets and the Marr conjecture We now specialize Theorem 1 to the Ricker (Mexican hat) wavelet M(x) = P ∆G(x), which is clearly in . To apply the theorem we need to verify Con- ditions 1 and 2. 18 ##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##

label: paragraph

font: default

text: Ψ

label: caption

text: Figure

text: Graphs

text: of

text: ψ(x)

text: and

text: ψ

text: (x).

text: Note

text: ψ(x)

text: has

text: a

text: non-regular

text: zero

text: at

text: x

text: x

text: and

text: a

text: regular

text: zero

text: at

text: x

text: while

text: ψ

text: (x)

text: has

text: regular

text: zeros

text: at

text: x

text: and

text: x

text: and

text: no

text: non-regular

text: zeros

text: ber).

text: To

text: this

text: end,

text: solving

text: for

text: a

text: we

text: have

text: a

text: (x

text: e(

text: x

text: Now

text: substituting

text: for

text: a

text: in

text: and

text: solving

text: for

text: b:

text: b

text: (x

text: x

text: Now

text: substituting

text: in

text: and

text: rearranging,

text: (x

text: (x

text: If

text: x

text: were

text: algebraic

text: then

text: the

text: left

text: side

text: would

text: be

text: transcendental

text: (since

text: the

text: expo-

text: nentials

text: of

text: non-zero

text: algebraics

text: are

text: transcendental

text: by

text: the

text: Hermite-Lindemann

text: theorem)

text: while

text: the

text: right

text: side

text: would

text: be

text: algebraic,

text: which

text: would

text: give

text: a

text: contra-

text: diction.

text: Therefore

text: x

text: is

text: transcendental.

text: Ricker

text: wavelets

text: and

text: the

text: Marr

text: conjecture

text: We

text: now

text: specialize

text: Theorem

text: to

text: the

text: Ricker

text: (Mexican

text: hat)

text: wavelet

text: M(x)

text: P

text: ∆G(x),

text: which

text: is

text: clearly

text: in

text: To

text: apply

text: the

text: theorem

text: we

text: need

text: to

text: verify

text: Con-

text: ditions

text: and

text: ##LTFigure##

text: ##LTLine##

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text: 4 Geometry of Gaussian Edge Contours Having proven the Marr conjecture in one dimension (Corollary 3(a)), in the remainder of this work we ask whether this result can be extended to other sequences of scales and to functions that decay less rapidly than those P(cid:48) in . We therefore restrict our focus to one-dimensional Gaussian edges— γ ∗ ∗ ∈ R that is, zeros of f M , or equivalently, of ∆(f G )—for f L1( ). Our σ σ results are summarized in Corollary 3(b,c) above. To start, we give a characterization of the geometry of one-dimensional Gaussian edges, which we will later use in proving unique determination from sequences of bounded-scale edges. Since these edges are nodes of a heat equation solution, we will represent scale using the variable t = σ2 rather than σ. ∈ P(cid:48) ∩ R Given f L1( ), deﬁne γ ∗ √ F (x, t) = f G (x). t R× ∞ F is jointly analytic in both variables on the upper half-plane H = (0, ) + (e.g. [5, Theorem 10.3.1]). Both F and F = ∂2F satisfy the heat equation xx ∂x2 (2), and so are subject to the following maximum principle (e.g. [5, Theorem 15.3.1]): ≥ → R Theorem 10. For t > t 0, let s , s : [t , t ] be continuous functions 2 1 ∈ 1 2 1 2 with s (t) < s (t) for all t (t , t ]. Let D be the parabolic interior 1 2 1 2 { ∈ } D = (x, t) : t (t , t ], s (t) < x < s (t) , 1 2 1 2 Let u(x, t) satisfy the heat equation u = 1u on D¯, the closure of D. Then xx 2 t ¯ if the maximum (or minimum) value of u over D is achieved on D, u is ¯ constant on D. An immediate consequence of the maximum principle is that F has no xx isolated zeros, since such a zero would be a local extremum. Furthermore, since F is analytic, the resolution of analytic singularities in two real di- xx mensions [13, 12, 3] (or Puiseux series expansion, e.g. [4] or Theorem 4.2.11 of ∈ [19]) implies that for each zero (x , t ) H of F there must be a neighbor- ⊂ 0 0 + xx hood W H containing (x , t ) and a collection of injective real-analytic + 0 0 mappings 20 ##LTLine## ##LTLine## ##LTLine##

label: section

text: Geometry

text: of

text: Gaussian

text: Edge

text: Contours

text: Having

text: proven

text: the

text: Marr

text: conjecture

text: in

text: one

text: dimension

text: (Corollary

text: in

text: the

text: remainder

text: of

text: this

text: work

text: we

text: ask

text: whether

text: this

text: result

text: can

text: be

text: extended

text: to

text: other

text: sequences

text: of

text: scales

text: and

text: to

text: functions

text: that

text: decay

text: less

text: rapidly

text: than

text: those

text: in

text: We

text: therefore

text: restrict

text: our

text: focus

text: to

text: one-dimensional

text: Gaussian

text: edges—

text: γ

text: R

text: that

text: is,

text: zeros

text: of

text: f

text: M

text: or

text: equivalently,

text: of

text: ∆(f

text: G

text: )—for

text: f

text: Our

text: σ

text: σ

text: results

text: are

text: summarized

text: in

text: Corollary

text: above.

text: To

text: start,

text: we

text: give

text: a

text: characterization

text: of

text: the

text: geometry

text: of

text: one-dimensional

text: Gaussian

text: edges,

text: which

text: we

text: will

text: later

text: use

text: in

text: proving

text: unique

text: determination

text: from

text: sequences

text: of

text: bounded-scale

text: edges.

text: Since

text: these

text: edges

text: are

text: nodes

text: of

text: a

text: heat

text: equation

text: solution,

text: we

text: will

text: represent

text: scale

text: using

text: the

text: variable

text: t

text: σ2

text: rather

text: than

text: σ.

text: R

text: Given

text: f

text: deﬁne

text: γ

text: F

text: (x,

text: t)

text: f

text: G

text: (x).

text: t

text: R×

text: F

text: is

text: jointly

text: analytic

text: in

text: both

text: variables

text: on

text: the

text: upper

text: half-plane

text: H

text: (e.g.

text: Theorem

text: Both

text: F

text: and

text: F

text: satisfy

text: the

text: heat

text: equation

text: xx

text: and

text: so

text: are

text: subject

text: to

text: the

text: following

text: maximum

text: principle

text: (e.g.

text: Theorem

text: R

text: Theorem

text: For

text: t

text: t

text: let

text: s

text: s

text: [t

text: t

text: be

text: continuous

text: functions

text: with

text: s

text: (t)

text: s

text: (t)

text: for

text: all

text: t

text: (t

text: t

text: Let

text: D

text: be

text: the

text: parabolic

text: interior

text: D

text: (x,

text: t)

text: t

text: (t

text: t

text: s

text: (t)

text: x

text: s

text: (t)

text: Let

text: u(x,

text: t)

text: satisfy

text: the

text: heat

text: equation

text: u

text: on

text: D¯,

text: the

text: closure

text: of

text: D.

text: Then

text: xx

text: t

text: if

text: the

text: maximum

text: (or

text: minimum)

text: value

text: of

text: u

text: over

text: D

text: is

text: achieved

text: on

text: D,

text: u

text: is

text: constant

text: on

text: D.

text: An

text: immediate

text: consequence

text: of

text: the

text: maximum

text: principle

text: is

text: that

text: F

text: has

text: no

text: xx

text: isolated

text: zeros,

text: since

text: such

text: a

text: zero

text: would

text: be

text: a

text: local

text: extremum.

text: Furthermore,

text: since

text: F

text: is

text: analytic,

text: the

text: resolution

text: of

text: analytic

text: singularities

text: in

text: two

text: real

text: di-

text: xx

text: mensions

text: (or

text: Puiseux

text: series

text: expansion,

text: e.g.

text: or

text: Theorem

text: of

text: implies

text: that

text: for

text: each

text: zero

text: (x

text: t

text: H

text: of

text: F

text: there

text: must

text: be

text: a

text: neighbor-

text: xx

text: hood

text: W

text: H

text: containing

text: (x

text: t

text: and

text: a

text: collection

text: of

text: injective

text: real-analytic

text: mappings

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2c nΑn 1 1 1 1 (cid:90) (cid:45) (cid:45)Αn(cid:45)Βn (cid:45) (cid:43)Αn(cid:45)Βn (cid:45)Αn(cid:43)Βn (cid:43)Αn(cid:43)Βn − − Figure 3: The graph of h h − , which satsiﬁes ∆(h h − ) = c J . ∞ n n 1 n n 1 n αn,βn | − | Note that h (x) h − (x) dx = 4c α (1 + β ). For n = 1 this is the n n 1 n n n −∞ graph of h . h is constructed as an inﬁnite sum of functions of this form. 1 − As an induction step, we will choose real numbers c > 0, 0 < ( 1)n+1β < ∈ nR+1 n+1 α < 1, and 0 < σ < σ , such that if h L1( ) is deﬁned by n+1 n+1 n n+1 ∆h (x) =∆h (x) + c J , n+1 n n+1 αn+1,βn+1 then (cid:16) (cid:112) (cid:17)  (cid:16) (cid:17) (cid:112)  ∗ ± ∆(h G ) σ2 + 1 > 0 for k odd n+1 ∗ σk ± k ≤ ≤ (21) ∆(h G ) σ2 + 1 < 0 for k even, 1 k n + 1. n+1 σk k (cid:12) (cid:12) ≤ ≤ First, note that for any ﬁx(cid:12) ed σ (in particula(cid:12)r for σ = σ with 1 k n), k the quantity ∗ J G (x) αn+1,βn+1 σ is uniformly bounded over all choices of α and β and all x. Thus for n+1 n+1 ≤ c suﬃciently small, the desired relationships (21) hold for k n no matter n+1 the values of α and β . We choose c so that this property is satisﬁed n+1 n+1 n+1 and also c < c /2. n+1 n 26 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: nΑn

text: Figure

text: The

text: graph

text: of

text: h

text: h

text: which

text: satsiﬁes

text: ∆(h

text: h

text: c

text: J

text: n

text: n

text: n

text: n

text: n

text: αn,βn

text: Note

text: that

text: h

text: (x)

text: h

text: (x)

text: dx

text: α

text: β

text: For

text: n

text: this

text: is

text: the

text: n

text: n

text: n

text: n

text: n

text: graph

text: of

text: h

text: h

text: is

text: constructed

text: as

text: an

text: inﬁnite

text: sum

text: of

text: functions

text: of

text: this

text: form.

text: As

text: an

text: induction

text: step,

text: we

text: will

text: choose

text: real

text: numbers

text: c

text: α

text: and

text: σ

text: σ

text: such

text: that

text: if

text: h

text: is

text: deﬁned

text: by

text: n

text: ∆h

text: (x)

text: =∆h

text: (x)

text: c

text: J

text: n

text: then

text: ∆(h

text: G

text: σ2

text: for

text: k

text: odd

text: σk

text: k

text: ∆(h

text: G

text: σ2

text: for

text: k

text: even,

text: k

text: n

text: σk

text: k

text: First,

text: note

text: that

text: for

text: any

text: ed

text: σ

text: (in

text: for

text: σ

text: σ

text: with

text: k

text: n),

text: k

text: the

text: quantity

text: J

text: G

text: (x)

text: σ

text: is

text: uniformly

text: bounded

text: over

text: all

text: choices

text: of

text: α

text: and

text: β

text: and

text: all

text: x.

text: Thus

text: for

text: c

text: suﬃciently

text: small,

text: the

text: desired

text: relationships

text: hold

text: for

text: k

text: n

text: no

text: matter

text: the

text: values

text: of

text: α

text: and

text: β

text: We

text: choose

text: c

text: so

text: that

text: this

text: property

text: is

text: satisﬁed

text: and

text: also

text: c

text: c

text: n

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:  (cid:16) (cid:112) (cid:17) ≤ In this limit,the relationships (20) are preserved with in place of <: (cid:16) (cid:17) (cid:112)  ∗ ± ≥ ∆(h G ) σ2 + 1 0 for k odd σk k ∗ ± ≤ ∆(h G ) σ2 + 1 0 for k even. σk k This implies that the edge contours of G and G + h cross inﬁnitely often as → σ 0. Since the two edge contours of both G and G+h are symmetric about the σ-axis, the intersections of edge contours on each side of the σ-axis occur at the same σ-values. Thus the edges of G and G + h agree on an inﬁnite sequence of scales tending to zero. This proves Corollary 3(b). 6 Distributions with Finitely Many Moments We have shown that any one-dimensional function with exponential decay is uniquely determined by a sequence of scaled Gaussian edges, thus giving a suﬃcient condition for the Marr conjecture in one dimension. One can ask whether this result could be extended to functions that decay less rapidly— for example, functions with algebraic decay. Addressing this question re- quires a formal notion of distributions with only ﬁnitely many moments. To M that end, this section introduces the space of smooth test functions of | | MN(cid:48) asymptotic order x N or less, and its dual , whose elements are distri- N butions with moments through order N. We ﬁrst deﬁne these spaces, then M(cid:48) consider derivatives and antiderivatives of distributions in , and ﬁnally N we prove the existence and continuity of asymptotic moment expansions for such distributions. We consider only one-dimensional distributions, but the deﬁnitions and results presented here can readily be generalized to arbitrary dimensions. 6.1 Deﬁnitions M For any nonnegative integer N, let denote the space of smooth test R N ≥ functions ψ on such that, for each integer n 0, the seminorm || || | | − − | | ψ = sup (1 + x ) (N n) ψ(n)(x) (22) N,n ∈R x is ﬁnite. (These seminorms were ﬁrst introduced by Ho¨rmander [14] and are often used to deﬁne symbol classes of pseudodiﬀerential operators [31, e.g.].) 28 ##LTLine## ##LTLine##

text: In

text: this

text: limit,the

text: relationships

text: are

text: preserved

text: with

text: in

text: place

text: of

text: ∆(h

text: G

text: σ2

text: for

text: k

text: odd

text: σk

text: k

text: ∆(h

text: G

text: σ2

text: for

text: k

text: even.

text: σk

text: k

text: This

text: implies

text: that

text: the

text: edge

text: contours

text: of

text: G

text: and

text: G

text: h

text: cross

text: inﬁnitely

text: often

text: as

text: σ

text: Since

text: the

text: two

text: edge

text: contours

text: of

text: both

text: G

text: and

text: G+h

text: are

text: symmetric

text: about

text: the

text: σ-axis,

text: the

text: intersections

text: of

text: edge

text: contours

text: on

text: each

text: side

text: of

text: the

text: σ-axis

text: occur

text: at

text: the

text: same

text: σ-values.

text: Thus

text: the

text: edges

text: of

text: G

text: and

text: G

text: h

text: agree

text: on

text: an

text: inﬁnite

text: sequence

text: of

text: scales

text: tending

text: to

text: zero.

text: This

text: proves

text: Corollary

text: Distributions

text: with

text: Finitely

text: Many

text: Moments

text: We

text: have

text: shown

text: that

text: any

text: one-dimensional

text: function

text: with

text: exponential

text: decay

text: is

text: uniquely

text: determined

text: by

text: a

text: sequence

text: of

text: scaled

text: Gaussian

text: edges,

text: thus

text: giving

text: a

text: suﬃcient

text: condition

text: for

text: the

text: Marr

text: conjecture

text: in

text: one

text: dimension.

text: One

text: can

text: ask

text: whether

text: this

text: result

text: could

text: be

text: extended

text: to

text: functions

text: that

text: decay

text: less

text: rapidly—

text: for

text: example,

text: functions

text: with

text: algebraic

text: decay.

text: Addressing

text: this

text: question

text: re-

text: quires

text: a

text: formal

text: notion

text: of

text: distributions

text: with

text: only

text: ﬁnitely

text: many

text: moments.

text: To

text: M

text: that

text: end,

text: this

text: section

text: introduces

text: the

text: space

text: of

text: smooth

text: test

text: functions

text: of

text: asymptotic

text: order

text: x

text: N

text: or

text: less,

text: and

text: its

text: dual

text: whose

text: elements

text: are

text: distri-

text: N

text: butions

text: with

text: moments

text: through

text: order

text: N.

text: We

text: ﬁrst

text: deﬁne

text: these

text: spaces,

text: then

text: consider

text: derivatives

text: and

text: antiderivatives

text: of

text: distributions

text: in

text: and

text: ﬁnally

text: N

text: we

text: prove

text: the

text: existence

text: and

text: continuity

text: of

text: asymptotic

text: moment

text: expansions

text: for

text: such

text: distributions.

text: We

text: consider

text: only

text: one-dimensional

text: distributions,

text: but

text: the

text: deﬁnitions

text: and

text: results

text: presented

text: here

text: can

text: readily

text: be

text: generalized

text: to

text: arbitrary

text: dimensions.

text: Deﬁnitions

text: M

text: For

text: any

text: nonnegative

text: integer

text: N,

text: let

text: denote

text: the

text: space

text: of

text: smooth

text: test

text: R

text: N

text: functions

text: ψ

text: on

text: such

text: that,

text: for

text: each

text: integer

text: n

text: the

text: seminorm

text: ψ

text: sup

text: x

text: (N

text: n)

text: ψ(n)(x)

text: N,n

text: ∈R

text: x

text: is

text: ﬁnite.

text: (These

text: seminorms

text: were

text: ﬁrst

text: introduced

text: by

text: Ho¨rmander

text: and

text: are

text: often

text: used

text: to

text: deﬁne

text: symbol

text: classes

text: of

text: pseudodiﬀerential

text: operators

text: e.g.].)

text: ##LTLine##

text: ##LTLine##


text: ∗ is uniquely determined, up to a constant multiple, by the zero sets of f M { }∞ σi for any sequence of positive scales σ tending to inﬁnity. i i=1 This conjecture has remained open, although special cases have been proved [34, 7]. The corresponding statement for nondecaying functions was disproved by Meyer [27], who found distinct periodic functions whose Ricker wavelet transforms have identical zero sets at all scales. More generally, we can ask for minimal conditions on a general wavelet ψ allowing for such unique determination: Question. What conditions on a twice-diﬀerentiable function ψ are neces- sary and suﬃcient to imply that any function f, of suﬃciently rapid decay, is uniquely determined up to a constant multiple by (a) the zeros in (σ, x) ∗ of W f(σ, x) (b) the zero sets of f ψ , for any sequence of positive scales { }∞ σi σ tending to inﬁnity. i i=1 1.2 Results on unique determination Here we answer this question by ﬁnding conditions on f and ψ that are suﬃcient and the best of their type for such unique determination. We require that f be integrable and of exponential order—meaning that f belongs to a P(cid:48) class of exponentially decaying functions. We require that ψ belong to a Pγ class of smooth functions whose derivatives grow slower than exponentially, and satisfy the following: Genericity Condition. The regular zero set of any derivative of ﬁxed order n is not contained in the zero set of any other derivative of ﬁxed order m, ≥ for any n, m 0. A regular (transverse) zero of a function ψ is a point in all of whose neighborhoods ψ(x) takes both positive and negative values. By “derivative of ﬁxed order m”, we mean a linear combination of partial derivatives of ψ of order m, (i.e., a homogeneous linear diﬀerential operator of order m applied to ψ), modulo multiplication by a nonzero constant. As an example, the one-dimensional Gaussian wavelet G(x) fails this genericity condition, in that the regular zero set of G is empty and is therefore trivially contained in the zero set of G(n) for any n > 0. However its second derivative, the Ricker wavelet M(x), satisﬁes this condition, as we show in Section . Our main result can be stated as follows: 4

text: is

text: uniquely

text: determined,

text: up

text: to

text: a

text: constant

text: multiple,

text: by

text: the

text: zero

text: sets

text: of

text: f

text: M

text: σi

text: for

text: any

text: sequence

text: of

text: positive

text: scales

text: σ

text: tending

text: to

text: inﬁnity.

text: i

text: This

text: conjecture

text: has

text: remained

text: open,

text: although

text: special

text: cases

text: have

text: been

text: proved

text: The

text: corresponding

text: statement

text: for

text: nondecaying

text: functions

text: was

text: disproved

text: by

text: Meyer

text: who

text: found

text: distinct

text: periodic

text: functions

text: whose

text: Ricker

text: wavelet

text: transforms

text: have

text: identical

text: zero

text: sets

text: at

text: all

text: scales.

text: More

text: generally,

text: we

text: can

text: ask

text: for

text: minimal

text: conditions

text: on

text: a

text: general

text: wavelet

text: ψ

text: allowing

text: for

text: such

text: unique

text: determination:

text: Question.

text: What

text: conditions

text: on

text: a

text: twice-diﬀerentiable

text: function

text: ψ

text: are

text: neces-

text: sary

text: and

text: suﬃcient

text: to

text: imply

text: that

text: any

text: function

text: f,

text: of

text: suﬃciently

text: rapid

text: decay,

text: is

text: uniquely

text: determined

text: up

text: to

text: a

text: constant

text: multiple

text: by

text: (a)

text: the

text: zeros

text: in

text: (σ,

text: x)

text: of

text: W

text: f(σ,

text: x)

text: (b)

text: the

text: zero

text: sets

text: of

text: f

text: ψ

text: for

text: any

text: sequence

text: of

text: positive

text: scales

text: σi

text: σ

text: tending

text: to

text: inﬁnity.

text: i

text: Results

text: on

text: unique

text: determination

text: Here

text: we

text: answer

text: this

text: question

text: by

text: ﬁnding

text: conditions

text: on

text: f

text: and

text: ψ

text: that

text: are

text: suﬃcient

text: and

text: the

text: best

text: of

text: their

text: type

text: for

text: such

text: unique

text: determination.

text: We

text: require

text: that

text: f

text: be

text: integrable

text: and

text: of

text: exponential

text: order—meaning

text: that

text: f

text: belongs

text: to

text: a

text: class

text: of

text: exponentially

text: decaying

text: functions.

text: We

text: require

text: that

text: ψ

text: belong

text: to

text: a

text: Pγ

text: class

text: of

text: smooth

text: functions

text: whose

text: derivatives

text: grow

text: slower

text: than

text: exponentially,

text: and

text: satisfy

text: the

text: following:

text: Genericity

text: Condition.

text: The

text: regular

text: zero

text: set

text: of

text: any

text: derivative

text: of

text: ﬁxed

text: order

text: n

text: is

text: not

text: contained

text: in

text: the

text: zero

text: set

text: of

text: any

text: other

text: derivative

text: of

text: ﬁxed

text: order

text: m,

text: for

text: any

text: n,

text: m

text: A

text: regular

text: (transverse)

text: zero

text: of

text: a

text: function

text: ψ

text: is

text: a

text: point

text: in

text: all

text: of

text: whose

text: neighborhoods

text: ψ(x)

text: takes

text: both

text: positive

text: and

text: negative

text: values.

text: By

text: “derivative

text: of

text: ﬁxed

text: order

text: m”,

text: we

text: mean

text: a

text: linear

text: combination

text: of

text: partial

text: derivatives

text: of

text: ψ

text: of

text: order

text: m,

text: (i.e.,

text: a

text: homogeneous

text: linear

text: diﬀerential

text: operator

text: of

text: order

text: m

text: applied

text: to

text: ψ),

text: modulo

text: multiplication

text: by

text: a

text: nonzero

text: constant.

text: As

text: an

text: example,

text: the

text: one-dimensional

text: Gaussian

text: wavelet

text: G(x)

text: fails

text: this

text: genericity

text: condition,

text: in

text: that

text: the

text: regular

text: zero

text: set

text: of

text: G

text: is

text: empty

text: and

text: is

text: therefore

text: trivially

text: contained

text: in

text: the

text: zero

text: set

text: of

text: G(n)

text: for

text: any

text: n

text: However

text: its

text: second

text: derivative,

text: the

text: Ricker

text: wavelet

text: M(x),

text: satisﬁes

text: this

text: condition,

text: as

text: we

text: show

text: in

text: Section

text: Our

text: main

text: result

text: can

text: be

text: stated

text: as

text: follows:


text: Thus the unique determination result does not, in general, extend to func- tions with algebraic decay, leaving open only classes of functions with decay rates between exponential and algebraic, e.g. classes decaying as the log- − | | normal function f(x) = 1e (ln x )2 or faster. x Let N be a positive multiple of 4, and consider a positive symmetric ∈ R function h L1( ) satisfying the following conditions: ∈ M (cid:90) (i) For all φ − , N 1 | | ∞ φ(x) h(x) dx < . (30) R (cid:90) M(cid:48) (Thus h can be regarded as an element of − .) N 1 (cid:90) ∞ (ii) h has inﬁnite Nth moment: xNh(x) dx = . R (iii) h has second moment < 2: x2h(x) dx < 2. R (We note that the third condition can always be arranged by multiplying h by an appropriate constant.) ∈ Starting with any such h we will construct a pair of distributions f, g M(cid:48) − . We will show that f and g have exactly two persistent edge contours N 3 each, which are symmetric in the coordinate w = x/σ. We will further { } { } show that there is a sequence of pairs (w , σ ) ≥ , with w , σ > 0 and σ i i i 1 i i i (cid:0) (cid:1) (cid:0) (cid:1) increasing, such that (cid:0) (cid:1) (cid:0) (cid:1) ∗ ≥ ≥ ∗ ∆ f G (σ w ) 0 ∆ g G (σ w ) for i odd ∗ σi i i ≤ ≤ ∗ σi i i (31) ∆ f G (σ w ) 0 ∆ g G (σ w ) for i even. σi i i σi i i These statements together imply that edge contours of f and g intersect on a sequence of scales tending to inﬁnity. Finally, to obtain a violation of ∈ M Marr’s conjecture, we replace the distributions f, g − by the integrable ∗ ∗ N 3 functions f G and g G, whose edge contours are the same as those of f and g, but shifted by one unit in σ. The argument consists of two parts. The ﬁrst constructs of f and g, demonstrates the existence of two persistent, symmetric edge contours, and veriﬁes (31). The second part shows that f and g have no other persistent edge contours. Condition (iii) above will only be invoked in the second part. 36 ##LTLine##

text: Thus

text: the

text: unique

text: determination

text: result

text: does

text: not,

text: in

text: general,

text: extend

text: to

text: func-

text: tions

text: with

text: algebraic

text: decay,

text: leaving

text: open

text: only

text: classes

text: of

text: functions

text: with

text: decay

text: rates

text: between

text: exponential

text: and

text: algebraic,

text: e.g.

text: classes

text: decaying

text: as

text: the

text: log-

text: normal

text: function

text: f(x)

text: (ln

text: x

text: or

text: faster.

text: x

text: Let

text: N

text: be

text: a

text: positive

text: multiple

text: of

text: and

text: consider

text: a

text: positive

text: symmetric

text: R

text: function

text: h

text: satisfying

text: the

text: following

text: conditions:

label: list

text: M

label: equation

text: (i)

text: For

text: all

text: φ

text: N

text: φ(x)

text: h(x)

text: dx

text: R

text: (Thus

text: h

text: can

text: be

text: regarded

text: as

text: an

text: element

text: of

text: N

text: (ii)

text: h

text: has

text: inﬁnite

text: Nth

text: moment:

text: xNh(x)

text: dx

text: R

text: (iii)

text: h

text: has

text: second

text: moment

text: dx

text: R

text: (We

text: note

text: that

text: the

text: third

text: condition

text: can

text: always

text: be

text: arranged

text: by

text: multiplying

text: h

text: by

text: an

text: appropriate

text: constant.)

text: Starting

text: with

text: any

text: such

text: h

text: we

text: will

text: construct

text: a

text: pair

text: of

text: distributions

text: f,

text: g

text: We

text: will

text: show

text: that

text: f

text: and

text: g

text: have

text: exactly

text: two

text: persistent

text: edge

text: contours

text: N

text: each,

text: which

text: are

text: symmetric

text: in

text: the

text: coordinate

text: w

text: x/σ.

text: We

text: will

text: further

text: show

text: that

text: there

text: is

text: a

text: sequence

text: of

text: pairs

text: (w

text: σ

text: with

text: w

text: σ

text: and

text: σ

text: i

text: i

text: i

text: i

text: i

text: i

text: increasing,

text: such

text: that

text: f

text: G

text: (σ

text: w

text: g

text: G

text: (σ

text: w

text: for

text: i

text: odd

text: σi

text: i

text: i

text: σi

text: i

text: i

text: f

text: G

text: (σ

text: w

text: g

text: G

text: (σ

text: w

text: for

text: i

text: even.

text: σi

text: i

text: i

text: σi

text: i

text: i

text: These

text: statements

text: together

text: imply

text: that

text: edge

text: contours

text: of

text: f

text: and

text: g

text: intersect

text: on

text: a

text: sequence

text: of

text: scales

text: tending

text: to

text: inﬁnity.

text: Finally,

text: to

text: obtain

text: a

text: violation

text: of

text: M

text: Marr’s

text: conjecture,

text: we

text: replace

text: the

text: distributions

text: f,

text: g

text: by

text: the

text: integrable

text: N

text: functions

text: f

text: G

text: and

text: g

text: G,

text: whose

text: edge

text: contours

text: are

text: the

text: same

text: as

text: those

text: of

text: f

text: and

text: g,

text: but

text: shifted

text: by

text: one

text: unit

text: in

text: σ.

text: The

text: argument

text: consists

text: of

text: two

text: parts.

text: The

text: ﬁrst

text: constructs

text: of

text: f

text: and

text: g,

text: demonstrates

text: the

text: existence

text: of

text: two

text: persistent,

text: symmetric

text: edge

text: contours,

text: and

text: veriﬁes

text: The

text: second

text: part

text: shows

text: that

text: f

text: and

text: g

text: have

text: no

text: other

text: persistent

text: edge

text: contours.

text: Condition

text: (iii)

text: above

text: will

text: only

text: be

text: invoked

text: in

text: the

text: second

text: part.

text: ##LTLine##


text: ∈ P Theorem 1. Given ψ satisfying the above genericity condition, any ∈ P(cid:48) ∩ R f L1( d) is uniquely determined, up to a constant multiple, by the γ ∗ zero sets of its wavelet transform f ψ for any sequence of positive scales { }∞ σi σ tending to inﬁnity. i i=1 We will show that the conditions in this theorem are the best of their kind, in the following sense. First, the theorem fails if the exponential decay ∈ P condition f is weakened to algebraic decay (see Section ), although this γ leaves the conjecture open for the restricted set of functions f with decay −| | that is between algebraic and exponential, e.g. f(x) = e x 1/2 Second, if the genericity condition on the regular zeroes of the wavelet ψ (see above) fails weakly, then the theorem fails to hold (see Section ). Corollary 2. Given ψ and f as above, f is uniquely determined by the zero ∗ sets of its continuous wavelet transform f ψ for σ > 0, and more generally ∗ σi ∈ N its dyadic wavelet transform f ψ restricted to σ = 2i, i . σ In the case of the Ricker (Gaussian derivative) wavelet, we prove the following: Corollary 3. (Marr conjecture in one dimension) ∈ P(cid:48) ∩ R (a) Any f L1( d) is uniquely determined, up to a constant multiple, γ ∗ { }∞ by the zero sets of f M for any sequence of positive scales σ σi i i=1 with a nonzero limit point. { }∞ (b) This unique determination fails if the only limit point of σ is zero. i i=1 (c) This unique determination also fails if f is of algebraic rather than exponential order. For dimensions d > 1, the above theorem reduces the Marr conjecture to a statement about polynomial zeros. For any multiindex of nonegative integers α = (α , . . . , α ), we deﬁne the Laplace-Hermite polynomial L (x) 1 d α in x = (x , . . . , x ) by 1 d − | | ∆G(α)(x) = ( 1) α L (x)G(x). (1) α Above, the superscript (α) indicates a mixed partial derivative in the | | orders speciﬁed by α. Note that L is a polynomial of degree α + 2, where | | α α = α + . . . + α . We thus have: 1 d 5

text: P

text: Theorem

text: Given

text: ψ

text: satisfying

text: the

text: above

text: genericity

text: condition,

text: any

text: R

text: f

text: d)

text: is

text: uniquely

text: determined,

text: up

text: to

text: a

text: constant

text: multiple,

text: by

text: the

text: γ

text: zero

text: sets

text: of

text: its

text: wavelet

text: transform

text: f

text: ψ

text: for

text: any

text: sequence

text: of

text: positive

text: scales

text: σi

text: σ

text: tending

text: to

text: inﬁnity.

text: i

text: We

text: will

text: show

text: that

text: the

text: conditions

text: in

text: this

text: theorem

text: are

text: the

text: best

text: of

text: their

text: kind,

text: in

text: the

text: following

text: sense.

text: First,

text: the

text: theorem

text: fails

text: if

text: the

text: exponential

text: decay

text: P

text: condition

text: f

text: is

text: weakened

text: to

text: algebraic

text: decay

text: (see

text: Section

text: although

text: this

text: γ

text: leaves

text: the

text: conjecture

text: open

text: for

text: the

text: restricted

text: set

text: of

text: functions

text: f

text: with

text: decay

text: that

text: is

text: between

text: algebraic

text: and

text: exponential,

text: e.g.

text: f(x)

text: e

text: x

text: Second,

text: if

text: the

text: genericity

text: condition

text: on

text: the

text: regular

text: zeroes

text: of

text: the

text: wavelet

text: ψ

text: (see

text: above)

text: fails

text: weakly,

text: then

text: the

text: theorem

text: fails

text: to

text: hold

text: (see

text: Section

text: Corollary

text: Given

text: ψ

text: and

text: f

text: as

text: above,

text: f

text: is

text: uniquely

text: determined

text: by

text: the

text: zero

text: sets

text: of

text: its

text: continuous

text: wavelet

text: transform

text: f

text: ψ

text: for

text: σ

text: and

text: more

text: generally

text: σi

text: N

text: its

text: dyadic

text: wavelet

text: transform

text: f

text: ψ

text: restricted

text: to

text: σ

text: i

text: σ

text: In

text: the

text: case

text: of

text: the

text: Ricker

text: (Gaussian

text: derivative)

text: wavelet,

text: we

text: prove

text: the

text: following:

text: Corollary

text: (Marr

text: conjecture

text: in

text: one

text: dimension)

text: R

text: (a)

text: Any

text: f

text: d)

text: is

text: uniquely

text: determined,

text: up

text: to

text: a

text: constant

text: multiple,

text: γ

text: by

text: the

text: zero

text: sets

text: of

text: f

text: M

text: for

text: any

text: sequence

text: of

text: positive

text: scales

text: σ

text: σi

text: i

text: with

text: a

text: nonzero

text: limit

text: point.

text: (b)

text: This

text: unique

text: determination

text: fails

text: if

text: the

text: only

text: limit

text: point

text: of

text: σ

text: is

text: zero.

text: i

text: (c)

text: This

text: unique

text: determination

text: also

text: fails

text: if

text: f

text: is

text: of

text: algebraic

text: rather

text: than

text: exponential

text: order.

text: For

text: dimensions

text: d

text: the

text: above

text: theorem

text: reduces

text: the

text: Marr

text: conjecture

text: to

text: a

text: statement

text: about

text: polynomial

text: zeros.

text: For

text: any

text: multiindex

text: of

text: nonegative

text: integers

text: α

text: (α

text: α

text: we

text: deﬁne

text: the

text: Laplace-Hermite

text: polynomial

text: L

text: (x)

text: d

text: α

text: in

text: x

text: (x

text: x

text: by

text: d

text: ∆G(α)(x)

text: α

text: L

text: (x)G(x).

text: α

text: Above,

text: the

text: superscript

text: (α)

text: indicates

text: a

text: mixed

text: partial

text: derivative

text: in

text: the

text: orders

text: speciﬁed

text: by

text: α.

text: Note

text: that

text: L

text: is

text: a

text: polynomial

text: of

text: degree

text: α

text: where

text: α

text: α

text: α

text: α

text: We

text: thus

text: have:

text: d


text: We will need that, since Condition (b) on f and g holds for each k, k+1 k+1 µ (∆f ) > µ (∆g ) + 1 and µ (∆g ) > µ (∆f ) + 1, N k+1 N k N k+1 N k+1 (cid:90) (cid:90) it follows that ∞ ∞ xNh(x) dx = and xNh(x) dx = . (42) C D ≥ Additionally, since we required σ > σ + 1 for all i 1, the sequence { } i+1 i σ ≥ is increasing and diverges to positive inﬁnity. i i 1 7.2 Part 2: Non-existence of divergent edge contours For the second (ﬁnal) part of the argument, we must show that the persis- ± tent edge contours approaching w = 1 are the only persistent edge contours of f and g. We prove this for f, and the statement for g follows similarly. We begin by applying the moment expansion (Corollary 19 with M = 3) ∈ M(cid:48) ≥ ≤ − to the distribution ∆f − . (Recall N 4 and thus M N 1 for N 1 M = 3.) Since the moments of ∆f coincide with those of δ(2) through order − (cid:0) (cid:1) N 1, the quantity ∗ − σ3∆ f G (σw) H (w)G(w) σ 2 → ∞ converges to zero locally uniformly in w, as σ . Thus any persistent ± edge contours of f must either approach the roots w = 1 of H (w), or → ∞ 2 diverge in w as σ . We now show that the second case cannot occur. ⊂ Assume, to the contrary, that a persistent edge contour Z H of f ∞ → ∞ + diverges to (without loss of generality) + in w as σ . Deﬁne a mapping σ = s(x) so that for each x greater than or equal to some x > 0, ∈ 0 (cid:48) (x, s(x)) Z. (There is some freedom in this construction, since a line x = x may intersect Z multiple times.) By Corollary 14, local parameterizations of Z have no local maxima, so s(x) can be chosen to be monotone increasing in x. However, s(x) is not necessarily continuous—it may jump between { ∈ } branches of the set-valued function S(x) = σ : (x, σ) Z . ≥ For all x x , (x, s(x)) lies on an edge contour of f, so convolving (40) 0 44

text: We

text: will

text: need

text: that,

text: since

text: Condition

text: (b)

text: on

text: f

text: and

text: g

text: holds

text: for

text: each

text: k,

text: µ

text: (∆f

text: µ

text: (∆g

text: and

text: µ

text: (∆g

text: µ

text: (∆f

text: N

text: N

text: k

text: N

text: N

text: it

text: follows

text: that

text: xNh(x)

text: dx

text: and

text: xNh(x)

text: dx

text: C

text: D

text: Additionally,

text: since

text: we

text: required

text: σ

text: σ

text: for

text: all

text: i

text: the

text: sequence

text: i

text: σ

text: is

text: increasing

text: and

text: diverges

text: to

text: positive

text: inﬁnity.

text: i

text: i

text: Part

text: Non-existence

text: of

text: divergent

text: edge

text: contours

text: For

text: the

text: second

text: (ﬁnal)

text: part

text: of

text: the

text: argument,

text: we

text: must

text: show

text: that

text: the

text: persis-

text: tent

text: edge

text: contours

text: approaching

text: w

text: are

text: the

text: only

text: persistent

text: edge

text: contours

text: of

text: f

text: and

text: g.

text: We

text: prove

text: this

text: for

text: f,

text: and

text: the

text: statement

text: for

text: g

text: follows

text: similarly.

text: We

text: begin

text: by

text: applying

text: the

text: moment

text: expansion

text: (Corollary

text: with

text: M

text: to

text: the

text: distribution

text: ∆f

text: (Recall

text: N

text: and

text: thus

text: M

text: N

text: for

text: N

text: M

text: Since

text: the

text: moments

text: of

text: ∆f

text: coincide

text: with

text: those

text: of

text: through

text: order

text: N

text: the

text: quantity

text: σ3∆

text: f

text: G

text: (σw)

text: H

text: (w)G(w)

text: σ

text: converges

text: to

text: zero

text: locally

text: uniformly

text: in

text: w,

text: as

text: σ

text: Thus

text: any

text: persistent

text: edge

text: contours

text: of

text: f

text: must

text: either

text: approach

text: the

text: roots

text: w

text: of

text: H

text: (w),

text: or

text: diverge

text: in

text: w

text: as

text: σ

text: We

text: now

text: show

text: that

text: the

text: second

text: case

text: cannot

text: occur.

text: Assume,

text: to

text: the

text: contrary,

text: that

text: a

text: persistent

text: edge

text: contour

text: Z

text: H

text: of

text: f

text: diverges

text: to

text: (without

text: loss

text: of

text: generality)

text: in

text: w

text: as

text: σ

text: Deﬁne

text: a

text: mapping

text: σ

text: s(x)

text: so

text: that

text: for

text: each

text: x

text: greater

text: than

text: or

text: equal

text: to

text: some

text: x

text: (x,

text: s(x))

text: Z.

text: (There

text: is

text: some

text: freedom

text: in

text: this

text: construction,

text: since

text: a

text: line

text: x

text: x

text: may

text: intersect

text: Z

text: multiple

text: times.)

text: By

text: Corollary

text: local

text: parameterizations

text: of

text: Z

text: have

text: no

text: local

text: maxima,

text: so

text: s(x)

text: can

text: be

text: chosen

text: to

text: be

text: monotone

text: increasing

text: in

text: x.

text: However,

text: s(x)

text: is

text: not

text: necessarily

text: continuous—it

text: may

text: jump

text: between

text: branches

text: of

text: the

text: set-valued

text: function

text: S(x)

text: σ

text: (x,

text: σ)

text: Z

text: For

text: all

text: x

text: x

text: (x,

text: s(x))

text: lies

text: on

text: an

text: edge

text: contour

text: of

text: f,

text: so

text: convolving


text: 1.4 Results on the geometry of heat equation nodes Our work le(cid:40)ads to new results on the nodes of solutions to the heat equation initial value problem: ∈ −∞ ∞ ∈ ∞ F = 1F x ( , ), t [0, ) t 2 xx (2) F (x, 0) = f(x). The nodes (zeros) of F form algebraic curves which we call edge con- xx tours of f. We show that new edge contours do not appear as t increases, strengthening and complementing previous results [1, 34, 2, 15, 20, 32]: Theorem 7. For an integrable function f of exponential order and for pos- itive numbers t < t , the edge contours of f intersecting the line t = t are 1 2 2 a subset of those that intersect t = t . 1 We also obtain the following unique determination result: ∈ R Theorem 8. Let F be a solution(cid:90)to ((cid:90)2) for some initial condition f L1( ). If it is known that the second integral x y a(x) = f(z) dz dy −∞ −∞ is a function of exponential order, then f is uniquely determined by the zeros { }∞ of F (x, t ) for any sequence t of positive real numbers with a positive j j j=1 or inﬁnite limit point. 2 Moment Expansion Moment expansions represent functions (generally distributions) as series in derivatives | | ≡ ∂ α δ(α)(x) δ(x), ∂xα1 . . . ∂xαd 1 d based on the fact that these derivatives and the monomials ≡ xα xα1 . . . xαd 1 d form a biorthogonal system: 7 ##LTLine## ##LTLine##

text: Results

text: on

text: the

text: geometry

text: of

text: heat

text: equation

text: nodes

text: Our

text: work

text: to

text: new

text: results

text: on

text: the

text: nodes

text: of

text: solutions

text: to

text: the

text: heat

text: equation

text: initial

text: value

text: problem:

text: F

text: x

text: t

text: t

text: xx

text: F

text: (x,

text: f(x).

text: The

text: nodes

text: (zeros)

text: of

text: F

text: form

text: algebraic

text: curves

text: which

text: we

text: call

text: edge

text: con-

text: xx

text: tours

text: of

text: f.

text: We

text: show

text: that

text: new

text: edge

text: contours

text: do

text: not

text: appear

text: as

text: t

text: increases,

text: strengthening

text: and

text: complementing

text: previous

text: results

text: Theorem

text: For

text: an

text: integrable

text: function

text: f

text: of

text: exponential

text: order

text: and

text: for

text: pos-

text: itive

text: numbers

text: t

text: t

text: the

text: edge

text: contours

text: of

text: f

text: intersecting

text: the

text: line

text: t

text: t

text: are

text: a

text: subset

text: of

text: those

text: that

text: intersect

text: t

text: t

text: We

text: also

text: obtain

text: the

text: following

text: unique

text: determination

text: result:

text: R

text: Theorem

text: Let

text: F

text: be

text: a

text: for

text: some

text: initial

text: condition

text: f

text: If

text: it

text: is

text: known

text: that

text: the

text: second

text: integral

text: x

text: y

text: a(x)

text: f(z)

text: dz

text: dy

text: is

text: a

text: function

text: of

text: exponential

text: order,

text: then

text: f

text: is

text: uniquely

text: determined

text: by

text: the

text: zeros

text: of

text: F

text: (x,

text: t

text: for

text: any

text: sequence

text: t

text: of

text: positive

text: real

text: numbers

text: with

text: a

text: positive

text: j

text: j

text: or

text: inﬁnite

text: limit

text: point.

text: Moment

text: Expansion

text: Moment

text: expansions

text: represent

text: functions

text: (generally

text: distributions)

text: as

text: series

text: in

text: derivatives

text: α

text: δ(α)(x)

text: δ(x),

text: ∂xα1

text: ∂xαd

text: d

text: based

text: on

text: the

text: fact

text: that

text: these

text: derivatives

text: and

text: the

text: monomials

text: xα

text: xαd

text: d

text: form

text: a

text: biorthogonal

text: system:

text: ##LTLine##

text: ##LTLine##


text: || || P with topology generated by the seminorms as γ and α both vary. γ,α is the space of smooth functions with slower than exponential growth. The P P P(cid:48) P(cid:48) dual spaces to and are denoted and respectively. Distributions P(cid:48) γ− | | γ P(cid:48) in decay as e γ x or faster, and while those in have exponential or γ P(cid:48) ⊂ P(cid:48) faster decay. Clearly for each γ > 0. γ ∈ P(cid:48) The asymptotic m(cid:88)oment expansion of a distribution f is [9, Theorem 4.3.1] − | | ∼ ( 1) α −| |− → ∞ f(σx) µ σ α d δ(α)(x) (σ ), α α! | |≥ α 0 where (cid:104) (cid:105) µ = f(x), xα α ∈ P is the αth moment of f. This expansion holds in that for any ψ and ≥ (cid:88) N 0, (cid:104) (cid:105) µ −| |− O − − − → ∞ f(σx), ψ(x) = α σ α d ψ(α)(0)+ (σ N d 1) (σ ). (5) α! ≤| |≤ 0 α N (cid:12) (cid:12) The above asym(cid:12)ptotic expansion is equivalent to the follo(cid:12)wing equation ≥ (cid:12) (cid:88) (cid:12) for all N 0: (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:104) (cid:105) − µ −| |− lim σN+d f(σx), ψ(x) α σ α d ψ(α)(0) = 0. →∞ σ ≤| |≤ α! 0 α N ≤ Note that for polynomial ψ of degree N, the two sides of (5) coincide (without the error term) according to the biorthogonality relation (3). The ∈ P moment expansion (5) for general ψ is an asymptotic version of this biorthogonality relation. 2.2 Local uniform convergence of convolved moment expansions Here we prove the continuity result, Theorem 5 from Section , which we state here in a more precise form: ∈ P(cid:48) ∈ P ≥   Theorem 5. For all f , ψ , and N 0, the σ-indexed family of (cid:88) functions   − | | (cid:55)−→ ∗ − ( 1) α −| |− w σN+d f ψ (σw) µ σ α d ψ(α)(w) σ α α! ≤| |≤ 0 α N 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P

text: with

text: topology

text: generated

text: by

text: the

text: seminorms

text: as

text: γ

text: and

text: α

text: both

text: vary.

text: γ,α

text: is

text: the

text: space

text: of

text: smooth

text: functions

text: with

text: slower

text: than

text: exponential

text: growth.

text: The

text: P

text: P

text: dual

text: spaces

text: to

text: and

text: are

text: denoted

text: and

text: respectively.

text: Distributions

text: γ−

text: γ

text: in

text: decay

text: as

text: e

text: γ

text: x

text: or

text: faster,

text: and

text: while

text: those

text: in

text: have

text: exponential

text: or

text: γ

text: faster

text: decay.

text: Clearly

text: for

text: each

text: γ

text: γ

text: The

text: asymptotic

text: expansion

text: of

text: a

text: distribution

text: f

text: is

text: Theorem

text: α

text: f(σx)

text: µ

text: σ

text: α

text: d

text: δ(α)(x)

text: (σ

text: α

text: α!

text: α

text: where

text: µ

text: f(x),

text: xα

text: α

text: P

text: is

text: the

text: αth

text: moment

text: of

text: f.

text: This

text: expansion

text: holds

text: in

text: that

text: for

text: any

text: ψ

text: and

text: N

text: µ

text: O

text: f(σx),

text: ψ(x)

text: α

text: σ

text: α

text: d

text: (σ

text: N

text: d

text: (σ

text: α!

text: α

text: N

text: The

text: above

text: expansion

text: is

text: equivalent

text: to

text: the

text: equation

text: for

text: all

text: N

text: µ

text: lim

text: σN+d

text: f(σx),

text: ψ(x)

text: α

text: σ

text: α

text: d

text: σ

text: α!

text: α

text: N

text: Note

text: that

text: for

text: polynomial

text: ψ

text: of

text: degree

text: N,

text: the

text: two

text: sides

text: of

text: coincide

text: (without

text: the

text: error

text: term)

text: according

text: to

text: the

text: biorthogonality

text: relation

text: The

text: P

text: moment

text: expansion

text: for

text: general

text: ψ

text: is

text: an

text: asymptotic

text: version

text: of

text: this

text: biorthogonality

text: relation.

text: Local

text: uniform

text: convergence

text: of

text: convolved

text: moment

text: expansions

text: Here

text: we

text: prove

text: the

text: continuity

text: result,

text: Theorem

text: from

text: Section

text: which

text: we

text: state

text: here

text: in

text: a

text: more

text: precise

text: form:

text: P

text: Theorem

text: For

text: all

text: f

text: ψ

text: and

text: N

text: the

text: σ-indexed

text: family

text: of

text: functions

text: α

text: w

text: σN+d

text: f

text: ψ

text: (σw)

text: µ

text: σ

text: α

text: d

text: ψ(α)(w)

text: σ

text: α

text: α!

text: α

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 Ford et al. 3.5 e 3.0 d u plit 2.5 m 2.0 A e r 1.5 u s o 1.0 Cl 0.5 300 350 400 450 500 550 600 650 700 750 sum of baseline lengths 0.06 ) d a 0.04 r ( e s a h 0.02 P e r u s 0.00 o Cl 0.02 150 200 250 300 350 400 450 500 550 sum of baseline lengths Fig. 15.— Closure phase (in radians) and closure amplitude for a 7.5 mag point source (black) and noisy observations of 585 mas horizontal bars surrounding a 7.5 mag point source at 1, 2, and 5 mag integrated ﬂux contrasts (red, green and blue, respectively). Total baseline length is in arbitrary units. Black circles represent CP (CA) values for a noiseless point source, which depart from theoretical expectations due to pixellation. Error bars are based on measured variance of CP (CA) for each baseline in 10 noisy point source observations, and show 1σ departures corrected for the number of independent CPs and CAs. C.P. 6128 Succ. Centre-ville, Qc, H3C 3J7, Canada and Se´bastien Parmentier, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794 Subject headings: galaxies: active – galaxies: Seyfert – quasars: general – instrumentation: inter- ferometers – techniques: high angular resolution – space vehicles: instruments – methods: data analysis – techniques: image processing – accretion – accretion disks 4. EXAMPLE AGN TARGETS space-based NRM missions, spanning a range of activity types and black hole masses (M ). Figure 16 shows Table 1 lists targets for JWST-NIRISS AMI and future BH the science topics within reach of the various missions. Since the size of an AGN central engine is supposed to scale with M (Antonucci 1993; Urry & Padovani 1 Department of Astrophysics, American Museum of Natural BH 1995; Peterson et al. 2004), NIRISS AMI will probe dif- History, New York, NY 10024 2 CUNY Graduate Center, 365 5th Avenue, New York, NY ferent str−uctures around nearby AGNs with MBH span- 10016 ning 106 9M(cid:12). Column 8 in Table 1 lists the regions 3 Department of Physics & Astronomy, Stony Brook Univer- which can be imaged with JWST-NIRISS AMI, span- sity, Stony Brook, NY 11794 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## 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text: 16 Ford et al. TABLE 1 Galactic nuclei targets for NRM on Space Telescopes Name RA Dec. DL log(MBH) AGN Apparent JWST AFTA 16m ATLAST Magnitude 4.8µm 400nm 500nm (J2000) (J2000) (Mpc) class (for aperture) inner-outer inner-outer inner-outer (cid:48)(cid:48) (mag)( ) (pc) (pc) (pc) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ± NGC 4151 12 10 32.6 39 24 21 16.9 7.1 Sy 1 6.6 ± 1 (6) 6.2–49.9 1.41–11.3 0.3–2.11 NGC 4051 12 03 09.6 44 31 53 12.7 5.1 NLSy1 7.9 ± 1 (15) 4.7–37.5 1.06–8.5 0.2–1.63 NGC 3227 10 23 30.6 19 51 54 20.2 7.6 Sy1.5 8.0 1 (15) 7.5–59.7 1.7–13.5 0.32–2.59 ± M87 12 30 49.4 −12 23 28 22.3 9.8 Lr,j 7.5 ± 1 8.2–65.9 1.9–14.9 0.36–2.85 Cen A 13 25 27.6 43 01 09 11.0 7.7 Sy2 5.2 ± 0.2 (5) 4.1–32.5 0.92–7.3 0.18–1.41 M81 09 55 33.2 69 03 55 0.7 7.8 L 8.5 1 (3.5) 0.26–2.1 0.06–0.5 0.02–0.09 − ··· ··· Sgr A* 17 45 40.0 29 00 28 0.008 6.6 ± 609–4874AU 138–1100AU 26–205AU M31 00 42 44.3 41 16 09 0.8 7.6 L2 6.5 0.2 (15) 0.30–2.36 0.07–0.53 0.013–0.10 − ± NGC 1068 02 42 40.7 00 00 48 12.5 7.2 Sy 2 5∼.5 1.0(0.3) 4.6–37 1.04–8.33 0.2–1.6 NGC 4258 12 18 57.5 −47 18 14 9.0 7.6 Sy2,j ∼ 8.9 (7) 3.3–27 0.75–6.0 0.14–1.15 Circinus 14 13 09.9 −65 20 21 8.0 6.0 Sy1h ∼4–7 (0.4) 3.0–24 0.7–5.3 0.13–1.02 NGC 4945 13 05 27.5 49 28 06 11.1 6.1 Sy2 8.3 (14) 4.1–33 0.93–7.4 0.18–1.41 ∼ 3c 273 12 29 06.7 02 03 09 749 9.0 QSO 7.8 (4.6) 277–2212 62–499 12–96 Note. — Potential galactic nuclei targets for NRM observations with JWST-NIRISS AMI and proposed future missions, spanning a range of AGN types, black hole masses, distances and broad sky coverage. Column 1 is AGN name, column 4 is the luminosity distance (Mpc) listed in NED. Column 5 is the logarithm of the black hole mass (McKernan et al. 2010a). Column 6 is the AGN classiﬁcation(Sy=Seyfert, Lr=LINER, LL=low luminosity AGN,j=prominent jet, Bl=Blazar, QSO=quasar). Sy1h denotes a hidden Seyfert 1, visible only in polarized light and NLSy1 is a narrow line Sy1. Column 7 lists the range of observed L-band magnitudes of the AGNs (or as close as possible to 4.5µm), with (in brackets) the aperture size listed in arcseconds where available (from NED). Column 8 shows the distance scales (in pc exce∼pt for Sgr A*), that can be imaged in the AGN and galactic nuclei for JWST-NIRISS AMI and correspond to inner and outer working angles of 75-600 mas. Column 9 shows the corresponding distance scales that could be imaged by a 2.4 m AFTA operating at 400nm. Column 10 shows the distance scales that could be probed by a 16 m ATLAST mission operating at 500nm (for an 8 m ATLAST, multiply distances by two). K.E. Saavik Ford1,2, Barry McKernan1,2 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Anand Sivaramakrishnan1,3, Andre´ R. Martel, Anton Koekemoer, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 David Lafrenie`re 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Universit´e de Montr´eal, D´epartement de Physique C.P. 6128 Succ. Centre-ville, Qc, H3C 3J7, Canada and Se´bastien Parmentier, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794 Subject headings: galaxies: active – galaxies: Seyfert – quasars: general – instrumentation: inter- ferometers – techniques: high angular resolution – space vehicles: instruments – methods: data analysis – techniques: image processing – accretion – accretion disks 1 Department of Astrophysics, American Museum of Natural History, New York, NY 10024 2 CUNY Graduate Center, 365 5th Avenue, New York, NY 10016 3 Department of Physics & Astronomy, Stony Brook Univer- sity, Stony Brook, NY 11794 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ford

text: et

text: al.

text: TABLE

text: Galactic

text: nuclei

text: targets

text: for

text: NRM

text: on

text: Space

text: Telescopes

text: Name

text: RA

text: Dec.

text: DL

text: log(MBH)

text: AGN

text: Apparent

text: JWST

text: AFTA

text: ATLAST

text: Magnitude

text: (Mpc)

text: class

text: (for

text: aperture)

text: inner-outer

text: inner-outer

text: inner-outer

text: (mag)(

text: (pc)

text: (pc)

text: (pc)

text: NGC

text: Sy

text: NGC

text: NGC

text: Lr,j

text: Cen

text: A

text: L

text: Sgr

text: A*

text: NGC

text: Sy

text: NGC

text: Circinus

text: ∼4–7

text: NGC

text: QSO

text: Note.

text: Potential

text: galactic

text: nuclei

text: targets

text: for

text: NRM

text: observations

text: with

text: JWST-NIRISS

text: AMI

text: and

text: proposed

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text: missions,

text: spanning

text: a

text: range

text: of

text: AGN

text: types,

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text: and

text: broad

text: sky

text: coverage.

text: Column

text: is

text: AGN

text: name,

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text: is

text: the

text: luminosity

text: distance

text: (Mpc)

text: listed

text: in

text: NED.

text: Column

text: is

text: the

text: logarithm

text: of

text: the

text: black

text: hole

text: mass

text: (McKernan

text: et

text: al.

text: Column

text: is

text: the

text: AGN

text: classiﬁcation(Sy=Seyfert,

text: Lr=LINER,

text: LL=low

text: luminosity

text: AGN,j=prominent

text: jet,

text: Bl=Blazar,

text: QSO=quasar).

text: denotes

text: a

text: hidden

text: Seyfert

text: visible

text: only

text: in

text: polarized

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text: and

text: is

text: a

text: narrow

text: line

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text: the

text: range

text: of

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text: the

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text: brackets)

text: the

text: aperture

text: size

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text: in

text: arcseconds

text: where

text: available

text: (from

text: NED).

text: Column

text: shows

text: the

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text: scales

text: (in

text: pc

text: exce∼pt

text: for

text: Sgr

text: A*),

text: that

text: can

text: be

text: imaged

text: in

text: the

text: AGN

text: and

text: galactic

text: nuclei

text: for

text: JWST-NIRISS

text: AMI

text: and

text: correspond

text: to

text: inner

text: and

text: outer

text: working

text: angles

text: of

text: mas.

text: Column

text: shows

text: the

text: corresponding

text: distance

text: scales

text: that

text: could

text: be

text: imaged

text: by

text: a

text: m

text: AFTA

text: operating

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text: Column

text: shows

text: the

text: distance

text: scales

text: that

text: could

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text: by

text: a

text: m

text: ATLAST

text: mission

text: operating

text: at

text: (for

text: an

text: m

text: ATLAST,

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text: distances

text: by

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text: K.E.

text: Saavik

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text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

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text: York,

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text: Anand

text: Andre´

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text: C.P.

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text: Subject

text: headings:

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text: galaxies:

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text: quasars:

text: general

text: instrumentation:

text: inter-

text: ferometers

text: techniques:

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text: angular

text: resolution

text: space

text: vehicles:

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text: methods:

text: data

text: analysis

text: techniques:

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text: processing

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text: Department

text: of

text: Astrophysics,

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text: Museum

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text: Natural

text: History,

text: New

text: York,

text: NY

text: CUNY

text: Graduate

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text: Avenue,

text: New

text: York,

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text: Department

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text: Astronomy,

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