Wavelet transforms on Gelfand Shilov spaces and concrete examples

R E S E A R C H

Open Access

Wavelet transforms on Gelfand-Shilov

spaces and concrete examples

Naohiro Fukuda

_{, Tamotu Kinoshita}

_{and Kazuhisa Yoshino}

*_{Correspondence:}

[email protected]

2_{Institute of Mathematics, Tsukuba}

University, Tsukuba, Ibaraki 305-8571, Japan

Full list of author information is available at the end of the article

Abstract

In this paper, we study the continuity properties of wavelet transforms in the Gelfand-Shilov spaces with the use of a vanishing moment condition. Moreover, we also compute the Fourier transforms and the wavelet transforms of concrete functions in the Gelfand-Shilov spaces.

MSC: 42C10; 46F05; 65T60

Keywords: wavelet transform; Gelfand-Shilov space; continuity properties

1 Introduction

In recent years, the wavelet transform has been shown to be a successful tool in signal pro-cessing applications such as data compression and fast computations. The wavelet trans-form off ∈L_{(R) with respect to the analyzed waveletψ∈L}_{(R) satisfying the admissible} conditionCψ:=

R| ˆψ(ξ)|/|ξ|dξ<∞is defined by

Wψf(a,b) = 

Cψ

R

f(x)ψa,b(x)dx,

where

ψa,b(x) =



√ aψ

x–b

a

(a> ,b∈R)

(see [, ] for example). The inverse wavelet transform ofF∈L_{(R+×R) with respect to} the analyzed waveletψ∈L(R) is defined by

MψF(x) = 

Cψ

R+

R

F(a,b)ψa,b(x)

db da

a (x∈R).

For the time-frequency analysis, we are concerned with better localization in both time and frequency spaces from a point of view of the uncertainty principle. For the well-balanced localization, it would be suitable to consider the Schwartz spaceSregarded as the space of functions which have arbitrary polynomial decay and whose Fourier trans-forms also have arbitrary polynomial decay (see []). For instance, the typical Mexican hat wavelet belongs to spaces of more rapidly decreasing and more regular functions inS. In

this article we focus on Gelfand-Shilov spaces of functions which have sub-exponential de-cay and whose Fourier transforms also have sub-exponential dede-cay. For positive constants μ,νandhsuch thatμ+ν≥, we define the Banach Gelfand-Shilov space

Sμ ν,h(R) =

f ∈S;xα∂_xβf(x)_L∞_(R)≤Chα+βα!νβ!μfor allα,β∈N

with the norm

f_Sμ

ν,h(R)=_α,βsup_∈N

xα_∂β

xf(x)L∞(R)

hα+β_α!ν_β!μ ,

and the (non-Banach) Gelfand-Shilov spaceSμ ν(R)

Sμ

ν(R) =ind lim

h>S μ ν,h(R)

with the inductive limit topology. The Gelfand-Shilov spaces were originally introduced in [] and []. As well explained in [] and [], the Gelfand-Shilov spaces are better adapted to the study of the problems of partial differential equations for which the solutions sub-exponentially decay at infinity.

Remark . Restricting functions with Fourier transforms supported in the right half-plane, we may also define the Banach progressive Gelfand-Shilov space

Sμ,+ ν,h(R) =

f∈S_ν,μ_h(R);suppfˆ⊂[,∞) ,

and the (non-Banach) progressive Gelfand-Shilov space

Sμ,+ ν (R) =

f∈Sνμ(R);suppfˆ⊂[,∞) .

Such spaces can be considered in dealing with analytic signals as the Hardy spaceH_(R) (see []). If the analyzed waveletψ belongs to the progressive Gelfand-Shilov space,ψˆ smoothly tends to zero and also has vanishing moments. For example, the Bessel wavelet ψ(x) defined byψ(ξˆ ) =e–ξ–ξ–_{forξ>  andψ(ξ}ˆ _{) =  forξ≤ belongs toS},+

 (R). Actually, we know thatψ(x) =_π√

–ixK( √

 –ix), whereKis the first modified Bessel function of the second kind (see []).

For the discrete wavelet case requiring strong additional conditions, the Meyer wavelets or the Gevrey wavelets constructed as in [] belong to the Gelfand-Shilov spaces. As for the continuous wavelet transform requiring only the admissible condition, there are many possibilities to choose the analyzed wavelet. Boundedness results in a generalized Sobolev space, Besov space and Lizorkin-Triebel space are given in []. As forψ∈Sμ

ν(R) andψ∈

Sμ,+

2 Results

To state our results, we also introduce the following lemma.

Lemma . There exist C> and h> such that

eh|x|/ν_f

L∞(R)+e

h|ξ|/μ_fˆ

L∞(R)≤C

if and only if f ∈S_ν,μ_h(R).

For the proof refer to [, ], etc. Taking Lemma . into account, we denote another Banach Gelfand-Shilov space combining with the infinite vanishing moment condition

|ˆf(ξ)| ≤Ce–h|ξ|–/δ_,

Sμ,δ_ν,h(R) =f ∈S;eh|x|/νf_L∞_(R)+ehmax{|ξ|

/μ_,|ξ|–/δ_}ˆ

f_L∞_(R)<∞ .

We remark thatS_ν,μ_h(R) corresponds toSμ,δ_ν,h(R) withδ=∞, i.e.,

Sμ,∞_ν,h (R) =f ∈S;eh|x|/νf_L∞_(R)+eh|ξ| /μ_ˆ

f_L∞_(R)<∞ .

Remark . In particular, when fˆ(ξ) is just equal toe–h|ξ|–/δ

, it belongs to the Gevrey space of orderδ+ . So,νcan be taken asν≥δ+ .

Remark . We easily obtainehmax{|ξ|/μ,|ξ|–/δ}_≤eh(|ξ|/μ+|ξ|–/δ)_{. On the other hand, the} weight can be estimated from below as

ehmax{|ξ|/μ,|ξ|–/δ}≥eh(|ξ|/μ+|ξ|–/δ–)≥ceh(|ξ|/μ+|ξ|–/δ). ()

Therefore, we find thateh(|ξ|/μ+|ξ|–/δ)ˆfL∞(R)∼ ehmax{|ξ|

/μ_,|ξ|–/δ_}ˆ

fL∞(R).

Then we prove the following.

Theorem . Letμ,ν,h,handδbe positive constants such thatμ+ν≥,h<h.Define that d(λ) =λ(λ– )–+/λ.Then,for the wavelet transform Wψwith the waveletψ∈Sμ,δ_ν,h(R),

the following estimates hold:

forf ∈S∞,∞_ν,h (R)

(i) e

h|b/(a+)|/ν

a/_{+ } Wψf

L∞(R+×R)

≤Ceh|x|/νf_L∞_(R) ifν> ,

(i) eh–/ν|b/(a+)|/νWψf_L∞_(R+×R)≤Ce

h|x|/ν_f

L∞(R) if <ν≤,

forf ∈Sμ,∞_∞,h(R)

(ii) a

/_ehd(δ/μ+)/μ_a–/(μ+δ)

a+  Wψf

L∞(R+×R)

≤Ceh|ξ|/μfˆ_L∞_(R) ifμ> ,

(ii) a–/eh–/μd(δ/μ+)/μa–/(μ+δ)Wψf_L∞_(R+×R)≤Ce

h|ξ|/μˆ_f L∞(R)

forf ∈Sμ,δ_∞,h(R)

(iii) a

/_ehd(δ/μ+)/μ_(max{a,a–_})/(μ+δ)

a+  Wψf

L∞(R+×R) ≤Cehmax{|ξ|/μ,|ξ|–/δ}fˆ_L∞_(R) ifμ> ,

(iii) a–/eh–/μd(δ/μ+)/μ(max{a,a–})/(μ+δ)Wψf_L∞_(R+×R)

≤Cehmax{|ξ|/μ,|ξ|–/δ}fˆ_L∞_(R) if <μ≤.

Remark . We find that d(λ) is strictly greater than  for λ>  sinced(λ) = –(λ– )–+/λ_λ–_{log(λ– ), andd(λ) has the maximum at the pointλ=  and lim}

λ→+d(λ) =

lim_λ→∞d(λ) = .

Remark . This work is motivated by [] wheref andψ are allowed to take each dif-ferent value of parameters ν, μ and have infinite vanishing moments, more precisely vanishing moments of arbitrary polynomial order. Therefore, we have restricted our-selves to the case of f and ψ under the common parameters ν, μ, and have derived the above estimates with δ (concerning vanishing moments of sub-exponential order). For instance, eh(|b|/(ν+μ–)+(max{a,a–})/(ν+μ–))_W

ψfL∞ is estimated by [] with ρ=s=μ, ρ =ν, τ =τ=ν+μ–  andt= ν+μ–  (in the case off andψ under the com-mon parameters ν andμ). If one considers small a>  and takesν =δ+  (see Re-mark .), a similar estimate as (ii) holds since (max{a,a–_})/(ν+μ–)_(max{a,a–_})/(μ+δ)_∼

a–/(μ+δ)_{. Thanks to the additional condition of sub-exponential order, (i) for small}

a>  can become better sinceμ+ν≥ and |b|/(ν+μ–)=|b|/(ν+(μ+ν–))≤ |b|/ν ∼ |b/ (a+ )|/ν_.

Considering the study of the continuity properties in [, ] and [], we introduce spaces of functions inaandbwhich correspond to the Gelfand-Shilov spaces of functions inx

andξ sincea∼/|ξ|andb∼xafter wavelet transforms. Therefore, we shall define the following weighted L∞(R+×R) space which is a subspace ofL_{(R+×R) as far ashis} positive:

V_ν,μ,δ_h(R+×R) =

F∈L(R+×R);ehmax{|b/(a+)|

/ν_,a/μ_,a–/δ_}

F_L∞_(R+×R)<∞ .

We remark that ifμ=∞,

V_ν,∞,δ_h (R+×R) =F∈L(R+×R);ehmax{|b/(a+)|/ν,a–/δ}F_L∞_(R+×R)<∞ .

By (i) and (ii), we have

eh_a|b//(a+)|_{+ }/ν +

a/ehd(δ/μ+)/μa–/(μ+δ) a+ 

Wψf

L∞(R+×R)

≤Ceh|x|/νf_L∞_(R)+eh|ξ| /μ_ˆ

f_L∞_(R)

The weight function can be estimated from below as

eh|b/(a+)|/ν

a/_{+ } +

a/_ehd(δ/μ+)/μa–/(μ+δ)

a+  ≥ce

hmax{|b/(a+)|/ν_,a–/(μ+δ)_}

,

here we used Remark . also to eliminate the terma/_{. Therefore, by Theorem ., we} can also get the following continuity properties.

Corollary . Letμ,ν,h andδbe constants such thatμ> ,ν> ,h> andδ> .Then,

for the waveletψ∈S_ν,μ,δ_h(R),the wavelet transform S_ν,μ,∞_h (R)f →Wψf ∈V_ν,∞,μ+δ_h (R+×R)

is continuous.In particular,when f also satisfies the infinite vanishing moment condition,

the wavelet transform S_∞,μ,δ_h(R)f →Wψf∈V_∞,μ+δ,μ+δ_h (R+×R)is continuous.

In Section  we shall discuss the optimality of our boundedness results in Gelfand-Shilov spaces.

3 Proof of Theorem 2.4

At first, we introduce the following lemma.

Lemma . It holds that forα,β≥

α/θ+β/θ≥

⎧ ⎨ ⎩

–/θ_(α+β)/θ _{if <θ≤,}

(α+β)/θ+ ( – /θ)min{α/θ,β/θ} ifθ> .

Remark . The latter inequality is given in [] and [], which also shows multiplication algebras for the Gevrey-modulation spaces.

Proof of Lemma. We shall suppose thatα≥β>  since the proof is trivial whenα=  orβ= . Puttingγ :=α/β(≥), we may show

γ/θ+ ≥

⎧ ⎨ ⎩

–/θ_{(γ+ )}/θ _{if  <θ≤,}

(γ + )/θ+  – /θ ifθ> .

This follows from

min

γ≥

γ/θ+  (γ+ )/θ

= –/θ if  <θ≤,

and

min

γ≥

γ/θ+  – (γ+ )/θ =  – /θ ifθ> .

In the proofs of theorems, · denotes theL∞norm on R or R+×R. We shall consider the following cases.

•Case ofν>  anda≥) From the definition of the wavelet transform we get

eh|b/max{,a}|/νWψf(a,b)≤

Ceh|x|/νf a/

R

Lemma . withα=|b/a–x/a|,β=|x/a|gives

|x|/ν+x–b

a

/ν– b

max{,a} /ν

=b

a– x a /ν+x

a /ν–b

a

/ν+|x|/ν

 –

a /ν

≥b

a– x a +x

a

/ν+ – /νminb

a– x a /ν,x

a

/ν–b

a /ν

+|x|/ν

 –

a /ν

≥ – /νminb

a– x a /ν,x

a

/ν+|x|/ν

 –

a /ν,

here we used (|b/a–x/a|+|x/a|)/ν≥ |b/a|/ν. Therefore, puttingD:={x∈R;|b/a–x/a|<

|x/a|}, we have

eh|b/max{,a}|/νWψf(a,b)≤

Ceh|x|/ν_f

a/

D

e–h(–/ν)|b/a–x/a|/νdx

+Ce

h|x|/ν

f a/

R\D

e–h(–/ν)|x/a|/νdx

≤Ca/eh|x|/νf

R

e–h(–/ν)|x|/νdx

≤Ca/eh|x|/νf.

•Case ofν>  and  <a< ) Lemma . withα=|b–x|,β=|x|gives

|x|/ν+x–b

a

/ν– b

max{,a} /ν

=|b–x|/ν+|x|/ν–|b|/ν+|b–x|/ν a /ν– 

≥|b–x|+|x|/ν+ – /νmin|b–x|/ν,|x|/ν –|b|/ν+|b–x|/ν a /ν– 

≥ – /νmin|b–x|/ν,|x|/ν +|b–x|/ν a /ν– 

,

here we used (|b–x|+|x|)/ν_{≥ |b|}/ν_{. Therefore, putting}

I:=x∈R;|b–x|<  ,

we find that

I

e–h|b–x|/ν{|/a|/ν–}dx=

b+

b–

e–h|(b–x)/a|/ν{–a/ν}dx

=a /a

–/a

for  <a<  and have

eh|b/max{,a}|/νWψf(a,b)

≤Ceh|x| /ν

f a/

I

e–h|b–x|/ν{|/a|/ν–}dx

+Ce

h|x|/ν

f a/

R\I

e–h{(–/ν)min{|b–x|/ν,|x|/ν}+|/a|/ν–}dx

≤Mh,νa/eh|x|

/ν

f

+M_h,νeh|x|/νf

R

e–h(–/ν)min{|b–x|/ν,|x|/ν}dx

≤Ceh|x|/νf,

here we used _a/ e–h{|/a| /ν_–}

≤M_h,νfor  <a< .

Thus, sincemax{,a} ≤a+  andmax{,a/_{} ∼a}/_{+ , it follows that}

eh_a|b//(a+)|_{+ }/νWψf

≤Ceh|x|/νf.

•Case of  <ν≤ anda≥) Forh>h> , we get

eh–/ν|b/max{,a}|/νWψf(a,b)

≤Ceh|x| /ν

f a/

×

∞

–∞e

–(h–h){|x|/ν_+|(x–b)/a|/ν_}–h{|x|/ν_+|(x–b)/a|/ν_––/ν_{|b/max{,a}|}/ν_}

dx.

Lemma . withα=|b/a–x/a|,β=|x/a|gives

|x|/ν+x–b

a

/ν– –/ν b

max{,a} /ν

=b

a– x a /ν+x

a

/ν– –/νb

a

/ν+|x|/ν

 –

a /ν

≥–/νb

a– x a +x

a

/ν– –/νb

a

/ν+|x|/ν

 –

a /ν

≥ |x|/ν

 –

a /ν.

Therefore, we have

eh–/ν|b/max{,a}|/νWψf(a,b)≤

Ceh|x|/ν

f a/

∞

–∞e

–(h–h){|x|/ν+|(x–b)/a|/ν}_dx

≤ Ceh|x| /ν

f a/

∞

–∞e

–(h–h)|x|/ν_dx

•Case of  <ν≤ and  <a< ) Lemma . withα=|b–x|,β=|x|gives

|x|/ν+x–b

a

/ν– –/ν b

max{,a} /ν

=|b–x|/ν+|x|/ν– –/ν|b|/ν+|b–x|/ν a /ν– 

≥–/ν|b–x|+|x|/ν– –/ν|b|/ν+|b–x|/ν a /ν– 

≥ |b–x|/ν a /ν– 

.

Therefore, we have

eh–/ν|b/max{,a}|/νWψf(a,b)≤

Ceh|x|/ν_f

a/

∞

–∞e

–(h–h){|x|/ν_+|(x–b)/a|/ν_}

dx

≤ Ceh|x| /ν

f a/

_∞

–∞e

–(h–h)|(x–b)/a|/ν

dx

≤Ceh|x|/νf.

Thus, sincemax{,a} ≤a+ , it follows that

eh–/ν|b/(a+)|/νWψf≤Ceh|x|

/ν

f.

•Case ofμ> ) Letμ:=μ/(μ+δ). By Parseval’s theorem, the wavelet transform can be rewritten as

Wψf(a,b) =

a Cψ

R

ˆ

f(ξ)e–ibξ_ψˆ_{(aξ)dξ. ()}

Since

ehmax{|aξ|/μ,|aξ|–/δ}≥eh(|aξ|/μ+|aξ|–/δ–)≥ceh(|aξ|/μ+|aξ|–/δ)

similarly as (), we see that

e–ibξψ(ˆ aξ)≤Ce–h(|aξ|/μ+|aξ|–/δ).

Hence, we get

eh{d(δ/μ+)(+/a)μ}/μWψf(a,b)

≤Ca/eh|ξ|/μfˆ

∞

–∞e

–h{|ξ|/μ_+|aξ|/μ_+|aξ|–/δ_{–{d(δ/μ+)(+/a)}μ_}/μ_}

dξ.

Lemma . withα=|ξ|,β=|aξ|+|aξ|–μ/δ_gives

|ξ|/μ+|aξ|/μ+ 

|aξ|/δ –

d(δ/μ+ )

 +

a μ/μ

≥ |ξ|/μ+

|aξ|+ 

|aξ|μ/δ

/μ –

d(δ/μ+ )

 +

≥

|ξ|+|aξ|+ 

|aξ|μ/δ

/μ

+ – /μmin

|ξ|/μ,

|aξ|+ 

|aξ|μ/δ

/μ

–

d(δ/μ+ )

 +

a μ/μ

≥ – /μmin

|ξ|/μ,

|aξ|+ 

|aξ|μ/δ

/μ ,

here we used

min

ξ∈R

|ξ|+|aξ|+ 

|aξ|μ/δ

=d

δ μ+ 

 +

a μ

.

Therefore, putting

D:=ξ∈R;|ξ|/μ<|aξ|+|aξ|–μ/δ/μ ,

we have

eh{d(δ/μ+)(+/a)μ}/μWψf(a,b)

≤Ca/eh|ξ|/μfˆ

D

e–h(–/μ)|ξ|/μdξ

+Ca/eh|ξ|/μfˆ

R\D

e–h(–/μ)(|aξ|+|aξ|–μ/δ)/μdξ

≤ ⎧ ⎨ ⎩

Ca/eh|ξ|/μfˆ_Re–h(–/μ)|ξ|/μdξ ifa≥,

Ceh|ξ|/μ_fˆ

a/

Re–h(–

/μ_)|ξ|/μ

dξ if  <a< 

≤Cmaxa/,a–/ eh|ξ|/μfˆ.

Thus, since

eh{d(δ/μ+)(+/a)μ}/μ≥eh{d(δ/μ+)a–μ}/μ=ehd(δ/μ+)/μa–/(μ+δ)

andmax{a/_,a–/_}=a–/_{max{a, } ∼a}–/_{( +a), it follows that}

a/ehd(δ/μ+)_{ +a}/μa–/(μ+δ)Wψf

≤Ceh|ξ|/μfˆ.

•Case ofμ>  with the condition|ˆf(ξ)| ≤Ce–h|ξ|–/δ

) Letδ:=δ/(μ+δ). By () we get

eh{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μWψf(a,b)

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

×

∞

–∞e

Lemma . withα=|ξ|+|ξ|–μ/δ_{,β=|aξ|+|aξ|}–μ/δ_gives

|ξ|/μ+ 

|ξ|/δ +|aξ|

/μ₊ 

|aξ|/δ –

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

≥

|ξ|+ 

|ξ|μ/δ

/μ +

|aξ|+ 

|aξ|μ/δ

/μ –

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

≥

|ξ|+ 

|ξ|μ/δ +|aξ|+ 

|aξ|μ/δ

/μ

+ – /μmin

|ξ|+ 

|ξ|μ/δ

/μ ,

|aξ|+ 

|aξ|μ/δ

/μ

–

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

≥ – /μmin

|ξ|+ 

|ξ|μ/δ

/μ ,

|aξ|+ 

|aξ|μ/δ

/μ ,

here we used

min

ξ∈R

|ξ|+ 

|ξ|μ/δ +|aξ|+ 

|aξ|μ/δ

=d

δ μ+ 

( +a)μ

 + 

aμ/δ

δ .

Therefore, putting

D:=ξ∈R;|ξ|+|ξ|–μ/δ/μ<|aξ|+|aξ|–μ/δ/μ ,

we have

eh{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μWψf(a,b)

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

D

e–h(–/μ)(|ξ|+|ξ|–μ/δ)/μdξ

+Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

R\D

e–h(–/μ)(|aξ|+|aξ|–μ/δ)/μdξ

≤ ⎧ ⎨ ⎩

Ca/_eh(|ξ|/μ+|ξ|–/δ)_fˆ Re–h(–

/ν_)|ξ|/μ

dξ ifa≥,

Ceh(|ξ|/μ+|ξ|–/δ)ˆf a/

Re

–h(–/ν_)|ξ|/μ

dξ if  <a< 

≤Cmaxa/,a–/ eh(|ξ|/μ+|ξ|–/δ)fˆ.

Thus, since

( +a)μ

 + 

aμ/δ

δ

≥

max

a,

a μ

, ()

and

it follows that

a/ehd(δ/μ+)_{ +}/μ(max{_a a,a–})/(μ+δ)Wψf ≤Ceh(|ξ|

/μ_+|ξ|–/δ_)ˆ

f.

•Case of  <μ≤) Letk=max{/μ– , }– /δ. Forh>h> , we get

eh–/μ{d(δ/μ+)(+/a)μ}/μWψf(a,b)

≤Ca/eh|ξ|/μfˆ

×

∞

–∞e

–(h–h){|ξ|/μ+|aξ|/μ+|aξ|–/δ}

×e–h{|ξ|/μ+|aξ|/μ+|aξ|–/δ––/μ{d(δ/μ+)(+/a)μ}/μ}dξ.

We note that ifμ= , i.e.,  <μ< , there existsL>  such that

|aξ|+ 

|aξ|μ/δ

/μ

= 

|aξ|/δ

|aξ|+μ/δ+ /μ

≤ ⎧ ⎨ ⎩



|aξ|/δ((|aξ|+μ/δ)/μ+ /μ+L|aξ|/μ–) if|aξ| ≥, 

|aξ|/δ((|aξ|+μ/δ)/μ+ /μ+L|aξ|) if  <|aξ|< 

≤ 

|aξ|/δ

|aξ|+μ/δ/μ+ /μ+L|aξ|max{/μ–,}

=|aξ|/μ+ 

|aξ|/δ +L|aξ|

k_{. ()}

Ifμ= , () also holds withL= . Lemma . withα=|ξ|,β=|aξ|+ /|aξ|μ/δgives

|ξ|/μ+|aξ|/μ+ 

|aξ|/δ –  –/μ

d(δ/μ+ )

 +

a μ/μ

≥ |ξ|/μ+

|aξ|+ 

|aξ|μ/δ

/μ

–L|aξ|k– –/μ

d(δ/μ+ )

 +

a μ/μ

≥–/μ

|ξ|+|aξ|+ 

|aξ|μ/δ

/μ

–L|aξ|k– –/μ

d(δ/μ+ )

 +

a μ/μ

= –L|aξ|k,

here we used

min

ξ∈R

|ξ|+|aξ|+ 

|aξ|μ/δ

=d(δ/μ+ )

 +

a μ

.

There existR≥ >r>  independent ofa>  such that

Lh|aξ|k≤h–h|aξ|/μ+|aξ|–/δ for|aξ| ≥Ror|aξ|<r,

since –/δ<k=max{/μ– , }– /δ< /μ. Therefore, putting

we have

eh–/μ{d(δ/μ+)(+/a)μ}/μWψf(a,b)

≤Ca/eh|ξ|/μfˆ

R

e–(h–h)(|ξ|/μ+|aξ|/μ+|aξ|–/δ)+Lh|aξ|kdξ

≤Ca/eh|ξ|/μfˆ

R\I

e–(h–h)|ξ|/μdξ

+Ca/eh|ξ|/μfˆ

I

e–(h–h)(|ξ|/μ+|aξ|/μ+|aξ|–/δ)+Lh|aξ|kdξ

≤Ca/eh|ξ|/μfˆ

R

e–(h–h)|ξ|/μdξ

+Ca/eh|ξ|/μfˆ

R

e–(h–h)|ξ|/μ+Lhmax{Rk,rk}dξ

≤Ca/eh|ξ|/μfˆ.

Thus, it follows that

a–/eh–/μd(δ/μ+)/μa–/(μ+δ)Wψf≤a–/eh

_–/μ_{{d(δ/μ+)(+/a)}μ_}/μ

Wψf

≤Ceh|ξ|/μfˆ.

•Case of  <μ≤ with the condition|ˆf(ξ)| ≤Ce–h|ξ|–/δ

) Forh>h> , we get

eh–/μ{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μWψf(a,b)

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

×

∞

–∞e

–(h–h){|ξ|/μ_+|ξ|–/δ_+|aξ|/μ_+|aξ|–/δ_}

×e–h{|ξ|/μ+|ξ|–/δ+|aξ|/μ+|aξ|–/δ––/μ{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μ}dξ.

By () Lemma . withα=|ξ|+ /|ξ|μ/δ,β=|aξ|+ /|aξ|μ/δgives

|ξ|/μ+|ξ|–/δ+|aξ|/μ+ 

|aξ|/δ –  –/μ

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

≥

|ξ|+ 

|ξ|μ/δ

/μ +

|aξ|+ 

|aξ|μ/δ

/μ

–L|ξ|k+|aξ|k

– –/μ

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

≥–/μ

|ξ|+ 

|ξ|μ/δ +|aξ|+ 

|aξ|μ/δ

/μ

–L|ξ|k+|aξ|k

– –/μ

d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ/μ

here we used

min

ξ∈R

|ξ|+ 

|ξ|μ/δ +|aξ|+ 

|aξ|μ/δ

=d(δ/μ+ )( +a)μ

 + 

aμ/δ

δ .

There existR,R˜≥ >r,r˜>  independent ofa>  such that

Lh|aξ|k≤h–h|aξ|/μ+|aξ|–/δ for|aξ|>Ror|aξ|<r,

and forε>  satisfyingh>h+ε>h>  (e.g.,ε= (h–h)/)

Lh|ξ|k≤h–h–ε|ξ|/μ+|ξ|–/δ for|ξ|>R˜ or|ξ|<˜r,

since –/δ<k=max{/μ– , }– /δ< /μ. Therefore, putting

I:=ξ∈R;r/a≤ |ξ|<R/a and J:=ξ∈R;r˜≤ |ξ|<R˜ ,

we have

eh–/μ{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μWψf(a,b)

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

×

R

e–(h–h)(|ξ|/μ+|ξ|–/δ+|aξ|/μ+|aξ|–/δ)+Lh(|ξ|k_+|aξ|k₎

dξ

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

R\(I∪J)

e–ε(|ξ|/μ+|ξ|–/δ)dξ

+

I\J

e–ε(|ξ|/μ+|ξ|–/δ)–(h–h)(|aξ|/μ+|aξ|–/δ)+Lh|aξ|kdξ

+

J\I

e–(h–h)(|ξ|/μ+|ξ|–/δ)+Lh|ξ|kdξ

+

I∩J

e–(h–h)(|ξ|/μ+|ξ|–/δ+|aξ|/μ+|aξ|–/δ)+Lh(|ξ|k+|aξ|k)_dξ

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ

R

e–ε(|ξ|/μ+|ξ|–/δ)dξ

+

R

e–ε(|ξ|/μ+|ξ|–/δ)+Lhmax{Rk,rk}dξ

+

R

e–(h–h)(|ξ|/μ+|ξ|–/δ)+Lhmax{ ˜Rk,˜rk}dξ

+

R

e–(h–h)(|ξ|/μ+|ξ|–/δ)+Lhmax{Rk,R˜k,rk,˜rk}dξ

≤Ca/eh(|ξ|/μ+|ξ|–/δ)fˆ.

Thus, by () it follows that forh>h> ,

a–/eh–/μd(δ/μ+)/μ(max{a,a–})/(μ+δ)Wψf

≤a–/eh–/μ{d(δ/μ+)(+a)μ(+/aμ/δ)δ}/μWψf≤Ceh(|ξ|

/μ_+|ξ|–/δ_)ˆ

f_.

4 Concrete examples

In this section, we introduce concrete examples according to whether the order of vanish-ing moments is finite or infinite.

•Case of finite vanishing moments) Let us consider the function and the wavelet

f(x) =sech(hx), ψ(x) = d

dxsech(hx).

In particular, whenh=π_, it holds thatfˆ(ξ) =f(ξ), and we also see thatψ(ξˆ ) =iξf(ξ) andψ∈S,∞_,h(R) with  <h<h. By the change of variablest=ehax, we have the wavelet

transform

Wψf(a,b) =

a Cψ

R

ψ(x)f(ax+b)dx

= –h

a Cψ

R

ehx_–e–hx

(ehx_+e–hx₎ ·



eh(ax+b)_+e–h(ax+b)dx

= – 

ehb_C

ψa

_∞



t/a_{– }

(t/a_{+ )}·

dt

ta–a_(t+e–hb₎ ,

where

Cψ=

R

| ˆψ(ξ)|

|ξ| dξ (= log).

Using the Hölder inequalityA+B≥_(p–)p–/pA/pB–/pwith

 < 

 +h_h <p±:=  ±_hmaxh_{,a} <



 –h_h <∞ forsgnb=± respectively,

we obtain the estimate (from above)

Wψf(a,b)

≤ 

ehb_Cψa

∞

 

t/a_{+ }·

dt

ta–a · p±

(p±–)–/p±t/p±e–hb(–/p±)

=(p±– )

–/p±_ehb(–/p±)

p±Cψa

∞

 

t/a_{+ }·

dt

ta–a+p_±

≤Chehb_√(–/p±)

a

_∞

 

t/a_{+ }·

dt

t–a± h

hmax{,a}

≤Chehb(–/p±) √

a



   + ·

dt

t–a±hmaxh{,a}

+

∞

 

t/a_{+ }·

dt

t–a± h

hmax{,a}

=Che

hb(–/p±)

√ a

  a±

h

hmax{,a}

+ 

 a ∓

h

hmax{,a}

=Che

hb(–/p±)

√ a

a

 – ( ha hmax{,a}) ≤Cha/e–h

_{|b|/max{,a}}

here we used the fact that

 –  a±

h

hmax{,a}< ,  +

 a±

h

hmax{,a}> .

Then (i)and (ii)in Theorem . become

(i) eh|b/(a+)|Wψf_L∞_(R+×R)≤Ceh|x|f_L∞_(R),

(ii) a–/ehWψf_L∞_(R+×R)≤Ceh|ξ|fˆ_L∞_(R).

From estimate () it is possible that this example is the near critical case of (i)and (ii) since|b/(a+ )| ∼ |b|/max{,a}.

Remark . If we consider the typical example of the Mexican hat wavelet

ψ(x) =  π/√_

 –xe–x/, ψˆ(ξ) = 

√

π π/√_ξ

_e–ξ/_,

we see thatψ∈S/,∞_/,h(R) with  <h<h= /. In particular, whenf(x) =e–x _/

∈S/,∞_/,h (R), the wavelet transform is computed as

Wψf(a,b) =

√π/_a/_(a_{–  –b}₎

Cψ(a+ )/

e–b/(a+).

Then (i)in Theorem . becomes

(i) eh–|b/(a+)|Wψf_L∞_(R+×R)≤Ce

h|x| f_L∞_(R).

The exponent –b_/(a_{+ ) is not a critical case of (i)with  <h<h=}

sinceh–|b/(a+ )|_∼b_/(a_{+ ). Therefore, we gave the new waveletψ(x) =} d

dxsech(hx)∈S

,∞

,h(R) with

 <h<h=π_.

•Case of infinite vanishing moments) Firstly we prove the following.

Proposition . The inverse Fourier transform of e–ξ–tξ–_{is given by}

F–_e–ξ–tξ– (x) = √

 ∞

n= (–t)n

n! F

 –n

 ,  , –

x 

, ()

whereF(a,b,z)is the confluent hypergeometric function of the first kind.

Remark . The change of variables also yields

F–_e–ξ/–tξ– (x) =

 π

∞



e–ξ/–tξ–cosxξdξ

=√ π

∞



e–ξ–(t/

√ )_ξ–

=√F–e–ξ–(t/

√ )ξ–

(√x)

= ∞

n=

(–√t)n

n! F

 –n

 ,  , –

x 

.

Proof of Proposition. Let us put

I(t,x) :=F–e–ξ–tξ–_{(x) =}

 π

∞



e–ξ–tξ–_cosxξdξ.

DifferentiatingI(t,x) inx, we have

∂xI(t,x) = –

 π

∞



e–ξ–tξ–ξsinxξdξ,

∂_xI(t,x) = –

 π

∞



e–ξ–tξ–ξcosxξdξ.

On the other hand, differentiatingIint, we also have

∂tI(t,x) = –

 πt

∞



e–ξ–tξ–ξ–cosxξdξ.

Moreover, the integration by parts yields

∂tI(t,x) = –

 πt

–

∞



e–tξ– e–ξξcosxξdξ

=

 πt

–

∞



e–ξ–tξ– – ξ_{cosxξ–xξsinxξ dξ.}

Thus, we see thatI(t,x) satisfies the partial differential equation

∂tI(t,x) = t–



I(t,x) +∂ 

xI(t,x) +

x

∂xI(t,x)

. ()

We may suppose thatx≥ sinceI(t,x) =

 π

∞  e

–ξ_–t_ξ–

cosxξdξ is an even function inx. Now we consider the pointx= √–y(y≤) and get forJ(t,y) :=I(t, √–y)

∂yJ(t,y) =

–

√_–

y(∂xI)(t,  √

–y), ∂_yJ(t,y) = –

y

∂_xI(t, √–y) – 

y∂yJ(t,y).

Therefore, by the change of variablesx= √–y, it holds that

∂tJ(t,y) = t–



J(t,y) –y∂ 

yJ(t,y) +

y–  

∂yJ(t,y)

.

To solve this partial differential equation, we shall use the method of separation of vari-ables. By puttingJ(t,y) =∞_n=Ln(t)Kn(y), we obtain

t∂tLn(t)

Ln(t)

=–y∂ 

yKn(y) – (_–y)∂yKn(y) +_Kn(y)

Kn(y)

We immediately see thatLn(t) =tλnLn(). It is known that

I(t, ) =

 π

∞



e–ξ–tξ–dξ

=√ e –t ≡√   +(–) ! t ₊(–) ! t _{+· · ·} . ()

We note that

I(t, ) =J(t, ) = ∞

n=

Ln(t)Kn() =

∞

n=

Ln(t),

here we may takeKn() =  for alln∈Nby choosing the suitableLn(t). Hence we see that

λn=n_ and

Ln(t) =tnLn() =



√

 (–)n

n! t

n_.

Meanwhile, the eigenvalue problem

–y∂_yKn(y) –

 –y

∂yKn(y) +

 Kn(y) =

n

Kn(y)

withKn() =  has

Kn(y) =F

 –n

 ,  ,y

.

Thus it follows that

J(t,y) = ∞

n=

Ln(t)Kn(y) =

∞ n=  √  (–)n

n! t

n

F

 –n

 ,  ,y

,

which gives

I(t,x) =√ 

∞

n=

tn(–)

n

n! F

 –n

 ,  , – x  . ()

We knew that I(t,x) is an even function in advance and supposed that x≥. The last representation also implies thatI(t,x) is an even function inx. So, () holds for allx∈R.

We have derived () by solving the partial differential equation. To avoid confusion, let us denote the solution represented as in () byI˜(t,x). It remains to show the uniqueness of

˜

I(t,x) =√ 

_∞

n=tn(–) n

n! F( –n

 ,  , –

x

) andI(t,x) =

 π

∞  e–ξ

_–t_ξ–

cosxξdξ except the case oft= . Instead ofI(t,x), we consider for (s,x)∈(,∞)×R

I(s,x)=I(√s,x)=

 π

_∞



for the differentiation with respect tos. Then, by Stirling’s formula, we obtain

∂_sm∂_xjI(s,x)≤

 π

∞



e–ξ/·e–sξ–ξ–m·e–ξ/ξj/dξ

≤Csup

η≥e

–sη_ηm_·sup

μ≥e –μ/_μj/

≤Ce–m

m s

m

·e–j/jj/

≤Cr_sm+jm!(j!)/ ≤Cr_sm+jm!j!.

This implies that I(s,x) is analytic for (s,x) ∈ [s,∞) × R with arbitrarily fixed

s> . Therefore, we see that I(t,x) =

 π

∞  e–ξ

_–t_ξ–

cosxξdξ is analytic for (t,x)∈

(,∞)×R.

Remark . ProbablyI(t,x) would be analytic also att= . ButI(s,x) (=I(√s,x)) loses the analyticity ats= . Indeed, we find thatI(√s, ) =√

e –√s_=√

{ + (–)

!

√

s+(–)_!s+· · · }.

The Taylor expansion around a pointt=T>  gives

I(t,x)

=

 π

∞



e–ξ–tξ–cosxξdξ

=

n≥,k≥

an,k(t–T)nxk,

sinceI(t,x) is an even function inx. By () we also get another Taylor expansion

˜

I(t,x) =√ 

∞

n=

(t–T) +T n(–)

n

n! F

 –n

 ,  , –

x



=

n≥,k≥ ˜

an,k(t–T)nxk.

ThenU(t,x) :=I(t,x) –I˜(t,x) =_{n≥,k≥}un,k(t–T)nxksatisfies

∂tU(t,x) = t–



U(t,x) +∂ 

xU(t,x) +

x

∂xU(t,x)

,

and by ()

U(t, )≡.

Therefore, we getun,=  for alln≥ and

n≥,k≥

nun,kt(t–T)n–xk=

n≥,k≥

(k+ )un,k+ (k+ )un,k+ (t–T)nxk, ()

here we used that

∂_xI=

n≥,k≥

k(k– )un,k(t–T)nxk–=

n≥,k≥

(k+ )(k+ )un,k+(t–T)nxk.

Moreover, the left-hand side of () is changed into

n≥,k≥

nun,kt(t–T)n–xk=

n≥,k≥

Thus, it holds that

nun,k+ (n+ )un+,kT= (k+ )

un,k+ (k+ )un,k+ .

Hence, whenun,=  for alln≥, we find thatun,=  for alln≥, and recursivelyun,k= 

for alln≥ andk≥. So, we have

U(t,x) =

n≥,k≥

un,k(t–T)nxk≡.

This concludes that I(t,x) (=

 π

∞  e–ξ

_–t_ξ–

cosxξdξ) must coincide with ˜I(t,x) (= 

√ 

∞

n=tn(–) n

n! F( –n

 ,  , –

x

)) for (t,x)∈(,∞)×R.

As an application of Proposition ., we can compute the Fourier transform and the wavelet transform of concrete functions in the Gelfand-Shilov spaces. So, now let us take

ˆ

ψ(ξ) =fˆ(ξ) =e–ξ–ξ–_{. We see thatψ,f ∈S}/,/

/,h (R) for someh>  sincee–ξ



givesμ= / and the Gevrey functione–ξ– _{givesδ= / andν= / by the Paley-Wiener theorem.} Then by () it follows that

Wψf(a,b) = 

a Cψ

∞



e–(+a)ξ–(+a–)ξ–cosbξdξ

= 

a Cψ( +a)

∞



e–ω–(+a–)(+a)ω–cos√ b

 +aωdω

=

a Cψ( +a)

∞

–∞e

–ω–(a+/a)ω–

ei

b √

+aω_dω

=

πa Cψ( +a)

F–_e–ω–(a+/a)ω–_√ b  +a

.

By the Paley-Wiener theorem, we find that for someρ> 

Wψf(a,b)≤Ce–ρ|b/ √

+a_|/

∼Ce–ρ|b/(+a)|/.

This implies that the order (i) in Theorem . is almost optimal with respect toaandb. Using Proposition . witht=  andt=a+ /a, we have the following.

Theorem . Letψˆ(ξ) =fˆ(ξ) =e–ξ–ξ–forξ= and= forξ= .Then

ψ(x) =f(x) =√ 

∞

n= (–)n

n! F

 –n

 ,  , –

x 

∈S_/,/,/_h (R)

for some h> ,and the wavelet transform is given by

Wψf(a,b) =

_π

a Cψ( +a)

∞

n=

{–(a+ /a)}n

n! F

 –n

 ,  , –

b ( +a₎

,

Remark . Especially whenb= , we also find

Wψf(a, )=

_π

a Cψ( +a)

e–(a+a)_{. ()}

Then (iii)in Theorem . becomes

a–/ehmax{a,a–}WψfL∞(R+×R)≤Ce

hmax{|ξ|,|ξ|–}ˆ_f L∞(R).

() implies thatmax{a,a–_{}in (iii)cannot be improved anymore sinceh∼ and}

hmaxa,a– ∼

a+

a

.

Remark . As introduced in Remark ., the Bessel waveletψ(x) satisfiesψˆ(ξ) =e–ξ–ξ– forξ>  andψˆ(ξ) =  forξ≤ belongs toS_,+(R). Hence, we also see that

ψ(x) = 

π√ –ixK( √

 –ix) + 

π√ +ixK( √

 +ix)

satisfiesψˆ(ξ) =e–|ξ|–|ξ|–forξ=  andψˆ(ξ) =  forξ=  belongs toS_(R) andS_,,_h(R) for someh> .

5 Conclusions

In this paper, we consider the Banach spaces of Gelfand-Shilov functions satisfying van-ishing moment conditions and study the wavelet transforms. Our contributions are as follows:

() We derived sharp estimates of the wavelet transforms which are useful for the time-frequency analysis, and stated the continuity properties of the wavelet transforms in Gelfand-Shilov spaces as a corollary.

() We computed the Fourier transforms and the wavelet transforms of concrete functions in the Gelfand-Shilov spaces. These examples show the optimality of estimates in Theorem ..

Appendix

Concerned with the inverse wavelet transform, we also get the following.

Theorem A. Letμ,ν,h,handδbe positive constants such thatμ+ν≥,h<h.Define that d(λ) =λ(λ– )–+/λ_{.Then,for the inverse wavelet transform M}

ψ with the wavelet ψ∈Sμ,δ_ν,h(R),the following estimates hold:for F∈V_ν,μ,δ_h(R+×R)

(iv) ehd(ν/μ+)|x|/(μ+ν)MψF_L∞_(R)

≤Ceh{|b/max{,a}|/ν+a/μ+a–/δ}F_L∞_(R+×R) ifν> ,

(iv) eh–/νd(ν/μ+)|x|/(μ+ν)MψF_L∞_(R)

(v) |ξ|

/_ehd(δ/μ+)/μ_{(max{|ξ|,|ξ|}–_})/(μ+δ)

|ξ|+  F[MψF]

L∞(R)

≤eh{|b/max{,a}|/ν+a/μ+a–/δ}F_L∞_(R+×R) ifμ> ,

(v) eh–/μd(δ/μ+)/μ(max{|ξ|,|ξ|–})/(μ+δ)F[MψF]_L∞_(R)

≤Ceh{|b/max{,a}|/ν+a/μ+a–/δ}F_L∞_(R+×R) if <μ≤.

The weight function of (iv) and (v) can be estimated as

eh{|b/max{,a}|/ν+a/μ+a–/δ}≤ehmax{|b/(a+)|/ν,a/μ,a–/δ},

and estimated from below as

ehd(ν/μ+)|x|/(μ+ν)≥ceh|x|/(μ+ν),

and

|ξ|/ehd(δ/μ+)/μ_{(max{|ξ|,|ξ|}–_})/(μ+δ)

|ξ|+  ≥

ceh{|ξ|/(μ+δ)_+|ξ|–/(μ+δ)_} |ξ|/_+|ξ|–/ ≥

c

e

h{|ξ|/(μ+δ)+|ξ|–/(μ+δ)}

in the same way with (). Therefore, by Theorem A., we can also get the following conti-nuity property.

Corollary A. Letμ,ν,h andδbe constants such thatμ> ,ν> ,h> andδ> .Then,

for the wavelet ψ∈Sμ,δ_ν,h(R),the inverse wavelet transform V_ν,μ,δ_h(R+×R)F→MψF∈

Sμ+δ,μ+δ_μ+ν,h (R)is continuous.

We shall only give a sketch of the proof of Theorem A..

•Case ofν> ) From the definition of the inverse wavelet transform we get

ehd(ν/μ+)|x|/(μ+ν)MψF(x)

≤Ceh{|b/max{,a}|/ν+a/μ+a–/δ}F

×

R+

R

a–/_e–h{|b/max{,a}|/ν+a/μ+a–/δ+|(x–b)/a|/ν–d(ν/μ+)|x|/(μ+ν)}_{db da.}

We shall use the Hölder inequalityA+B≥(Ap)/p_(Bq)/q_{withp=μ/ν+ ,q=ν/μ+ . If}

a≥, Lemma . withα=|x/a–b/a|,β=|b/a|gives

maxb{,a}

/ν+a/μ+a–/δ+x–b

a /ν–d

ν μ+ 

|x|/(μ+ν)

≥ – /νminx

a– b a /ν,b

a

/ν+x

a

/ν+a/μ–d

ν μ+ 

|x|/(μ+ν)+a–/δ

≥ – /νminx

a– b a /ν,b

a