Wavelet transforms on Gelfand-Shilov spaces and concrete examples

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.


Introduction
In recent years, the wavelet transform has been shown to be a successful tool in signal processing applications such as data compression and fast computations. The wavelet transform of f ∈ L  (R) with respect to the analyzed wavelet ψ ∈ L  (R) satisfying the admissible condition C ψ := R |ψ(ξ )|  /|ξ | dξ < ∞ is defined by (see [, ] for example). The inverse wavelet transform of F ∈ L  (R + × R) with respect to the analyzed wavelet ψ ∈ L  (R) is defined by 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 wellbalanced localization, it would be suitable to consider the Schwartz space S regarded as the space of functions which have arbitrary polynomial decay and whose Fourier transforms also have arbitrary polynomial decay (see []). For instance, the typical Mexican hat wavelet belongs to spaces of more rapidly decreasing and more regular functions in S. In this article we focus on Gelfand-Shilov spaces of functions which have sub-exponential decay and whose Fourier transforms also have sub-exponential decay. For positive constants μ, ν and h such that μ + ν ≥ , we define the Banach Gelfand-Shilov space with the norm Such spaces can be considered in dealing with analytic signals as the Hardy space H  (R) (see []). If the analyzed wavelet ψ belongs to the progressive Gelfand-Shilov space,ψ smoothly tends to zero and also has vanishing moments. For example, the Bessel wavelet ψ(x) defined byψ(ξ ) = e -ξ -ξ - for ξ >  andψ(ξ ) =  for ξ ≤  belongs to S ,+  (R). Actually, we know that ψ( where K  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 for ψ ∈ S μ ν (R) and ψ ∈ S μ,+ ν (R), [] and [] show the continuity properties of wavelet transforms by preparing spaces of functions in a and b, respectively. In this paper, we shall pay careful attention also to the parameter h as the radius of convergence in the analytic class and attempt to find a further detailed estimate with h. So, our purpose is to show the continuity properties in (strong) topologies of Banach Gelfand-Shilov spaces with the use of a vanishing moment condition and to give concrete examples which can indicate the optimality in Section .

Results
To state our results, we also introduce the following lemma.
Lemma . There exist C >  and h  >  such that For the proof refer to [, ], etc. Taking Lemma . into account, we denote another Banach Gelfand-Shilov space combining with the infinite vanishing moment condition Remark . In particular, whenf (ξ ) is just equal to e -h|ξ | -/δ , it belongs to the Gevrey space of order δ + . So, ν can be taken as ν ≥ δ + .

Remark . This work is motivated by []
where f and ψ are allowed to take each different value of parameters ν, μ and have infinite vanishing moments, more precisely vanishing moments of arbitrary polynomial order. Therefore, we have restricted ourselves 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).
Considering the study of the continuity properties in [, ] and [], we introduce spaces of functions in a and b which correspond to the Gelfand-Shilov spaces of functions in x and ξ since a ∼ /|ξ | and b ∼ x after wavelet transforms. Therefore, we shall define the following weighted L ∞ (R + × R) space which is a subspace of L  (R + × R) as far as h is positive: We remark that if μ = ∞, By (i) and (ii), we have The weight function can be estimated from below as e h|b/(a+)| /ν a / +  + a / e hd(δ/μ+) /μ a -/(μ+δ) a +  ≥ ce h max{|b/(a+)| /ν ,a -/(μ+δ) } , here we used Remark . also to eliminate the term a / . Therefore, by Theorem ., we can also get the following continuity properties.
Corollary . Let μ, ν, h and δ be constants such that μ > , ν > , h >  and δ > . Then, In particular, when f also satisfies the infinite vanishing moment condition, In Section  we shall discuss the optimality of our boundedness results in Gelfand-Shilov spaces.

Proof of Theorem 2.4
At first, we introduce the following lemma.

Lemma . It holds that for
Remark . The latter inequality is given in [] and [], which also shows multiplication algebras for the Gevrey-modulation spaces.
Proof of Lemma . We shall suppose that α ≥ β >  since the proof is trivial when α =  or β = . Putting γ := α/β ( ≥ ), we may show This follows from In the proofs of theorems, · denotes the L ∞ norm on R or R + × R. We shall consider the following cases.

Thus, by () it follows that for h > h > ,
This concludes the proof of Theorem ..

Concrete examples
In this section, we introduce concrete examples according to whether the order of vanishing moments is finite or infinite.
• Case of finite vanishing moments) Let us consider the function and the wavelet In particular, when h = π  , it holds thatf (ξ ) = f (ξ ), and we also see thatψ(ξ ) = iξ f (ξ ) and ψ ∈ S ,∞ ,h (R) with  < h < h. By the change of variables t = e hax , we have the wavelet transform

Using the Hölder inequality
we obtain the estimate (from above) 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 Then (i) in Theorem . becomes The exponent -b  /(a  + ) is not a critical case of (i) with  < h < h =   since h  - |b/(a + )|  ∼ b  /(a  + ). Therefore, we gave the new wavelet ψ( • Case of infinite vanishing moments) Firstly we prove the following. a, b, z) is the confluent hypergeometric function of the first kind.

Remark . The change of variables also yields
Proof of Proposition . Let us put Differentiating On the other hand, differentiating I in t, we also have Moreover, the integration by parts yields Thus, we see that I(t, x) satisfies the partial differential equation We may suppose that x ≥  since I(t, x) =  π ∞  e -ξ  -t  ξ - cos xξ dξ is an even function in x. Now we consider the point x =  √ -y (y ≤ ) and get for J(t, y) := I(t,  √ -y) Therefore, by the change of variables x =  √ -y, it holds that To solve this partial differential equation, we shall use the method of separation of variables. By putting J(t, y) = ∞ n= L n (t)K n (y), we obtain t∂ t L n (t) L n (t) = -y∂  y K n (y) -(  y)∂ y K n (y) +   K n (y) K n (y) =: λ n .
We immediately see that L n (t) = t λ n L n (). It is known that We note that here we may take K n () =  for all n ∈ N by choosing the suitable L n (t). Hence we see that λ n = n  and Meanwhile, the eigenvalue problem -y∂  y K n (y) - Thus it follows that which gives We knew that I(t, x) is an even function in advance and supposed that x ≥ . The last representation also implies that I(t, x) is an even function in x. So, () holds for all x ∈ R.
We have derived () by solving the partial differential equation. To avoid confusion, let us denote the solution represented as in () byĨ(t, x). It remains to show the uniqueness of ∞ n= t n (-) n n!  F  ( -n  ,   , -x   ) and I(t, x) =  π ∞  e -ξ  -t  ξ - cos xξ dξ except the case of t = . Instead of I(t, x), we consider for (s, x) ∈ (, ∞) × R e -ξ  -sξ - cos xξ dξ for the differentiation with respect to s. Then, by Stirling's formula, we obtain This implies that I(s, x) is analytic for (s, x) ∈ [s  , ∞) × R with arbitrarily fixed s  > . Therefore, we see that Remark . Probably I(t, x) would be analytic also at t = . But I(s, x) (= I( √ s, x)) loses the analyticity at s = . Indeed, we find that I( The Taylor expansion around a point t = T >  gives since I(t, x) is an even function in x. By () we also get another Taylor expansioñ Then U(t, x) := I(t, x) -Ĩ(t, x) = n≥,k≥ u n,k (t -T) n x k satisfies and by () Therefore, we get u n, =  for all n ≥  and Thus, it holds that nu n,k + (n + )u n+,k T = (k + ) u n,k + (k + )u n,k+ .
Hence, when u n, =  for all n ≥ , we find that u n, =  for all n ≥ , and recursively u n,k =  for all n ≥  and k ≥ . So, we have This concludes that 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 takê ψ(ξ ) =f (ξ ) = e -ξ  -ξ - . We see that ψ, f ∈ S /,/ /,h (R) for some h >  since e -ξ  gives μ = / and the Gevrey function e -ξ - gives δ = / and ν = / by the Paley-Wiener theorem. Then by () it follows that By the Paley-Wiener theorem, we find that for some ρ >  This implies that the order (i) in Theorem . is almost optimal with respect to a and b. Using Proposition . with t =  and t = a + /a, we have the following.
Theorem . Letψ(ξ ) =f (ξ ) = e -ξ  -ξ - for ξ =  and =  for ξ = . Then for some h > , and the wavelet transform is given by , a, b, z) is the confluent hypergeometric function of the first kind.
Remark . Especially when b = , we also find () implies that max{a, a - } in (iii) cannot be improved anymore since h ∼  and h  max a, a - ∼  a +  a .
• Case of  < ν ≤ ) This case can be shown similarly as the case of ν > .
• Case of μ > ) This case can be shown similarly as the case of μ >  with the condition |f (ξ )| ≤ Ce -|ξ | /δ for the wavelet transform by exchanging the roles of a and ξ .