Progressive Gelfand-Shilov Spaces and Wavelet Transforms

Abstract

We discuss progressive Gelfand-Shilov spaces consisting of analytic signals with almost exponential decay in time and frequency variables. It is shown that such signals enjoy an additional localization property. We define wavelet transform and inverse wavelet transform in (progressive) Gelfand-Shilov spaces and study their continuity properties. It is shown that with a slightly faster decay in domain we may control the decay of the wavelet transform independently in each variable.

1. Introduction

In this paper, we study a class of Gelfand-Shilov spaces and their closed subspaces of analytic signals which are almost exponentially localized in time and frequency variables. Since the support of Fourier transform of such signals is in [0,∞), those subspaces are called progressive Gelfand-Shilov spaces, see Definition 2.1. We use the well-known characterization of Gelfand-Shilov spaces, Theorem 2.6. Note that in contrast to the proofs given in [1–4] Theorem 2.6 can be proved without arguments based on the Fourier transform invariance properties, which is of interest when dealing with progressive Gelfand-Shilov spaces, see Remark 2.7. In particular, we show that progressive Gelfand-Shilov spaces enjoy an additional localization property, Proposition 2.8.

Note that the description of progressive Gelfand-Shilov type spaces given by almost exponential decay in time and frequency variables does not contain explicit regularity conditions.

The study of progressive Gelfand-Shilov spaces is motivated by the fact that they form closed subspaces of the Hardy space 𝐻²(ℝ) which consists of functions in 𝐿²(ℝ) whose Fourier transforms are supported by the right half-plane. It is well known that Hardy spaces 𝐻^𝑝(ℝ),1≤𝑝≤∞, play fundamental role in signal processing, cf. [5–7]. Analytic signal 𝐹, associated to a real-valued signal 𝑓∈𝐿²(ℝ) by 𝐹=𝑓+√−1𝐻(𝑓), where 𝐻(𝑓) is the Hilbert transform of 𝑓, belongs to 𝐻²(ℝ), which is the main motivation for the study of analytic signals in the framework of the Hardy space, in particular through the wavelet analysis, for example [5, 7–9].

For the study of (micro) local properties of analytic signals and of the corresponding dual spaces, analyzing wavelets should satisfy certain regularity and decay conditions. It is usually assumed that such wavelets have vanishing moments of certain order. In order to describe local properties of polynomial decay in time-frequency plane, it is sufficient to perform wavelet analysis in the Schwartz space of rapidly decreasing functions [9–11]. Analytic signals, however, might have better localization properties which cannot be captured within that framework. In the present paper, we show how progressive Gelfand-Shilov spaces can be used in this context. Our approach is a starting point for the study of wavelet transforms of (ultra) distributions of almost exponential growth at infinity which are not tempered distributions. This will be the subject of a separate paper. We remark that progressive functions cannot be exponentially localized simultaneously in time and in frequency domain.

In time-frequency analysis, weight functions are commonly used to quantify growth and decay conditions, for example [12]. From that point of view, it is natural to assume that results in certain spaces of ultra-distributions follow directly from the counterparts in the space of tempered distributions, by replacing polynomial type weights of the form ⟨⋅⟩^𝑠=(1+|⋅|²)^𝑠/2,𝑠>0, with subexponential weights of the form 𝑒^{𝑎|⋅|𝑏},𝑎>0, 0<𝑏<1. Although this is true in certain situations, and some results of the paper can be formulated to hold for both polynomial and subexponential weights, this cannot be done in full generality. The reason for this is different behavior of weights and their inverse functions with respect to convolution, translation, and dilation. Even in polynomial case, there is a difference between the power of the Haar module 𝑤_𝑠(𝑥,𝜉)=𝑤_𝑠(𝜉)=⟨𝜉⟩^𝑠,𝑠>0, and a weight of the form 𝑤_𝑠1,𝑠2(𝑥,𝜉)=⟨𝑥⟩^𝑠1⟨𝜉⟩^𝑠2, 𝑠₁,𝑠₂>0, (𝑥,𝜉)∈ℝ×(0,∞). The Haar module type weights are submultiplicative with respect to the action of the “𝑎𝑥+𝑏”-group and properties of the wavelet transform can be derived from the general theory of coorbit spaces, as it is done in [13–15]. This is not the case for 𝑤_𝑠1,𝑠2(𝑥,𝜉). Here, we give explicit calculations for subexponential weights and give explanations concerning more general situations whenever convenient.

Gelfand-Shilov spaces were introduced in [2] (and called there the spaces of type 𝑆) to study the uniqueness of the Cauchy problems of partial differential equations and since then they were observed both from theoretical (e.g. [1, 3, 4]) and applied point of view (e.g. [16–18]). For example, in [18] it is shown that these spaces can be used to construct a noncommutative quantum field theory.

Our results can be compared to [19] where one can find certain global type results. The analyzing wavelets in [19] may not satisfy the admissibility condition (see (4.24)) which is connected to the study of local regularity properties of analyzing functions, while our results describe both local and global behavior of wavelet transforms.

The paper is organized as follows. In Section 2, we recall the characterization of Gelfand-Shilov spaces, Theorem 2.6, and we introduce the progressive Gelfand-Shilov spaces and show a localization property of its elements, Proposition 2.8. In Section 3, we define the wavelet transform and study its decay properties, Propositions 3.2 and 3.3. This, together with the results from Section 2, give a continuity result, Theorem 3.5. Section 4 is devoted to the inverse wavelet transform, whose decay properties are studied in Propositions 4.1 and 4.2, and a continuity result and the inversion formula are given in Theorem 4.3. In the last section, we give a reformulation of results from Sections 3 and 4 under the assumption that both the analyzing wavelet and the analyzed function have slightly faster decay. This opens the possibility to control the decay of the wavelet transform in each variable separately. We believe that those results are of independent interest.

In order to simplify the exposition, we observe one-dimensional case only. Extension to higher dimensions is straightforward. Throughout the paper, we use standard notation. For example, (𝐿^𝑝(ℝ),‖⋅‖_𝑝),1≤𝑝≤∞, are Lebesgue spaces, ⟨𝑓,𝑔⟩ denotes the dual pairing of 𝑓 and 𝑔, and the Fourier transform of 𝑓∈𝐿¹(ℝ) is normalized to be ℱ𝑓(𝜉)=𝑓(𝜉)=_ℝ𝑒^{−2𝜋𝑖𝑥𝜉}𝑓(𝑥)𝑑𝑥,𝜉∈ℝ.(1.1) By 𝐷_𝑎 we denote the dilation 𝐷_𝑎(⋅)=(1/𝑎)𝑓(⋅/𝑎),𝑎∈ℝ₊=(0,∞). We adopt the following notation used in the study of pseudo-differential operators (cf. [17, 20]): ⟨⋅⟩=(1+|⋅|²)^1/2.

For reader's convenience, below we give the notation for spaces of test functions introduced in this paper and refer to corresponding definitions. Progressive Gelfand-Shilov spaces 𝑆^𝜈𝜇,+(ℝ) are introduced in Definition 2.1. The range of the wavelet transform over such spaces is contained in 𝑆^𝜇𝜈(ℝ×ℝ₊), see Definition 3.4, and in 𝑍^𝜇𝜈(ℝ×ℝ₊), see Definition 5.1, the latter being introduced for the analysis of decay of wavelet transform with respect to variables 𝑎 and 𝑏 separately, for example, Section 5.

2. Progressive Gelfand-Shilov Spaces

Let ℎ,𝜈, and 𝜇 be positive constants such that 𝜈+𝜇≥1. By 𝑆^𝜇𝜈,ℎ(ℝ) we denote the Banach space which consists of all 𝑓∈𝐶^∞(ℝ) such that the norm ‖𝑓‖_𝑆^𝜇𝜈,ℎ=sup_{𝛼,𝛽∈ℕ0}‖‖𝑥^𝛼𝑓^(𝛽)(𝑥)‖‖_𝐿∞ℎ^𝛼+𝛽𝛼!^𝜈𝛽!^𝜇(2.1) is finite.

The Gelfand-Shilov space 𝑆^𝜇𝜈(ℝ) is given by 𝑆^𝜇𝜈(ℝ)=_ℎ>0𝑆^𝜇𝜈,ℎ(ℝ),(2.2) and its topology is given by the inductive limit: 𝑆^𝜇𝜈(ℝ)=indlim_ℎ>0𝑆^𝜇𝜈,ℎ(ℝ).(2.3)

Definition 2.1. Let ℎ,𝜈, and 𝜇, be positive constants such that 𝜈+𝜇≥1. The space 𝑆^{𝜇,+𝜈,ℎ}(ℝ) consists of all 𝑓∈𝐶^∞(ℝ) such that supp𝑓∈[0,∞) and such that (2.1) is finite.
The progressive Gelfand-Shilov space 𝑆^𝜈𝜇,+(ℝ) is given by 𝑆^𝜈𝜇,+(ℝ)=_ℎ>0𝑆^{𝜇,+𝜈,ℎ}(ℝ),(2.4) and its topology is given by the inductive limit: 𝑆^𝜈𝜇,+(ℝ)=indlim_ℎ>0𝑆^{𝜇,+𝜈,ℎ}(ℝ).(2.5)

Obviously, 𝑆^𝜈𝜇,+(ℝ) is a closed subspace of 𝑆^𝜇𝜈(ℝ).

Remark 2.2. In the study of ultradistributions, it is of interest to observe projective limits of Banach spaces of test functions, apart from the inductive limits. Results of the present paper can be reformulated for progressive Gelfand-Shilov type spaces defined as projective limits of the spaces 𝑆^{𝜇,+𝜈,ℎ}(ℝ). For the projective limits of 𝑆^𝜇𝜈,ℎ(ℝ), ℎ>0, we refer to [21, 22].
Note also that the decay of ‖𝑥^𝛼𝑓^(𝛽)(𝑥)‖_𝐿^∞ can be controlled by some other choice of sequences {𝑀_𝛼},{𝑁_𝛽}, instead of the Gevrey sequences 𝑀_𝛼=𝛼!^𝜈 and 𝑁_𝛽=𝛽!^𝜇 used in (2.1). We refer to [1, 21–23] for the usual conditions on {𝑀_𝛼} and {𝑁_𝛽}.
For the sake of the clarity of the exposition, in this paper, we observe the inductive limits and the Gevrey sequences 𝛼!^𝜈 and 𝛽!^𝜇, which is not an essential restriction.

It can be shown that multiplication by 𝑥 and differentiation are continuous operations in 𝑆^𝜈𝜇,+(ℝ) and that the same is true for translation and dilation by a positive factor, for example [2, Chapter IV]. On the other hand, 𝑆^𝜈𝜇,+(ℝ) is not closed under modulations. This implies that Short-time Fourier transform, an important tool in local time-frequency analysis (see [24] for the definition and basic properties), is not appropriate for the analysis of 𝑆^𝜈𝜇,+(ℝ). For that reason, we cannot use the elegant arguments from for example [3, 4] in our analysis.

The following proposition can be used in the proof of Theorem 2.6, and it is of independent interest.

Proposition 2.3. Let 𝑠>0 and 𝑓∈𝐶^∞(ℝ). Then, the following conditions are equivalent. (a)There exists ℎ>0 such that sup_𝑥∈ℝ||𝑥^𝛼𝑓(𝑥)||≤𝐶ℎ^𝛼𝛼!^𝑠,∀𝛼∈ℕ₀,(2.6) for some 𝐶>0. (b)There exists 𝑘>0 such that sup_𝑥∈ℝ𝑒^{𝑘|𝑥|1/𝑠}||𝑓(𝑥)||<∞.(2.7)

The proof is omitted, since it can be found in [17, Proposition 6.1.5.], see also [2, Chapter IV]. Alternatively, Proposition 2.3 can be proved by the use of the following auxiliary result, see [25].

Lemma 2.4. Let 𝑡>0 and 𝜀>0. There exist 𝐶=𝐶(𝑡,𝜀)>0 such that 𝐶⁻¹𝑒^{(𝑡−𝜀)𝜂1/𝑡}≤^∞_𝑗=0𝜂^𝑗(𝑗!)^𝑡≤𝐶𝑒^{(𝑡+𝜀)𝜂1/𝑡},∀𝜂≥0.(2.8)

Remark 2.5. When 𝑠>0 and 𝑓∈𝐶^∞(ℝ) by Proposition 2.3, it directly follows that the following conditions are equivalent. (a)There exists ℎ>0 such that sup_𝜔>0|||𝜔+1𝜔^𝛼𝑓(𝜔)|||≤𝐶ℎ^𝛼𝛼!^𝑠,∀𝛼∈ℕ₀,(2.9) for some 𝐶>0.(b)There exists 𝑘>0 such that sup_𝜔>0𝑒^{𝑘(𝜔+1/𝜔)1/𝑠}||𝑓(𝜔)||<∞.(2.10)

Next, we recall the characterization of Gelfand-Shilov spaces.

Theorem 2.6. Let 𝜈 and 𝜇 be positive constants such that 𝜈+𝜇≥1 and let 𝑓∈𝐶^∞(ℝ). Then, the following conditions are equivalent,(a)𝑓∈𝑆^𝜇𝜈(ℝ). (b)There exist ℎ>0 and 𝐶>0 (which depend only on 𝑓) such that sup_𝑥||𝑥^𝛼𝑓(𝑥)||≤𝐶ℎ^𝛼𝛼!^𝜈,∀𝛼∈ℕ₀sup_𝑥||𝑓^(𝛽)(𝑥)||≤𝐶ℎ^𝛽𝛽!^𝜇,∀𝛽∈ℕ₀.(2.11)(c)There exist ℎ>0 and 𝐶>0 (which depend only on 𝑓) such that sup_𝑥||𝑥^𝛼𝑓(𝑥)||≤𝐶ℎ^𝛼𝛼!^𝜈,∀𝛼∈ℕ₀sup_𝜔||𝜔^𝛽𝑓(𝜔)||≤𝐶ℎ^𝛽𝛽!^𝜇,∀𝛽∈ℕ₀.(2.12)(d)There exist ℎ>0 and 𝐶>0 (which depend only on 𝑓) such that sup_𝜔|||𝜔_𝛼𝑓^(𝛽)(𝜔)|||≤𝐶ℎ^𝛼+𝛽𝛼!^𝜇𝛽!^𝜈,∀𝛽∈ℕ₀.(2.13)(e)There exist ℎ>0 and 𝐶>0 (which depend only on 𝑓) such that sup_𝑥𝑒^{ℎ|𝑥|1/𝜈}||𝑓(𝑥)||≤𝐶,sup_𝜔𝑒^{ℎ|𝜔|1/𝜇}||𝑓(𝜔)||≤𝐶.(2.14)

This theorem is well known, so we omits the proof. It can be proved as in [1, 3, 4] where arguments based on Fourier transform invariance of 𝑆^𝜇𝜈(ℝ)(ℱ𝑆^𝜇𝜈(ℝ)=𝑆^𝜈𝜇(ℝ)) are used. Another proof which follows from direct computations is given in [17, Theorem 6.1.6.], see also [21, 22].

Remark 2.7. By Theorem 2.6 immediately follows the characterization of progressive Gelfand-Shilov spaces 𝑆^𝜈𝜇,+(ℝ) where the supremum with respect to 𝜔 in Theorem 2.6(c), (d), and (e) is taken over 𝜔>0 instead of 𝜔∈ℝ.

Moreover, elements in 𝑆^𝜈𝜇,+(ℝ) satisfy an additional localization property, which is sometimes called "strip localization." We refer to [9] for the definition of polynomial strip localized progressive functions. Let us show that elements from 𝑆^𝜈𝜇,+(ℝ) are almost exponentially strip localized.

Proposition 2.8. Let 𝜈 and 𝜇 be positive constants such that 𝜈+𝜇≥1 and let 𝑓∈𝑆^𝜇𝜈(ℝ) satisfy supp𝑓∈[0,∞), that is, 𝑓∈𝑆^𝜈𝜇,+(ℝ). Then, sup_𝜔>0𝑒^{ℎ(𝜔+1/𝜔)1/𝑠}|𝑓(𝜔)|<∞, where 𝑠=max{𝜇,𝜈}.

Proof. Note that 𝑓∈𝑆^𝜈𝜇,+(ℝ) implies 𝑓^(𝑛)(0)=0,𝑛∈ℕ₀,  that is, _ℝ𝑥^𝑛𝑓(𝑥)𝑑𝑥=0,𝑛∈ℕ₀,(2.15) and 𝑓(𝜔)=𝑜(𝜔^𝑛) when 𝜔→0, for every 𝑛∈ℕ₀. By Taylor's formula, it follows that ||𝑓(𝜔)||=|||||^𝑛−1_𝑚₌₀𝑓^(𝑚)(0)𝑚!𝜔^𝑚+𝜔^𝑛𝑛!𝑛¹⁰≤𝜔(1−𝜃)^𝑛−1𝑓^(𝑛)(𝜃𝜔)𝑑𝜃_𝑛|||||𝑛!sup_𝜔>0|||𝑓^(𝑛)≤𝜔_𝑛(𝜔)|||𝑛¹⁰(1−𝜃)^𝑛−1𝑑𝜃𝐶ℎ^𝑛𝑛!^𝑠,(2.16) where we have used Theorem 2.6(d) and 𝑛∫¹⁰(1−𝜃)^𝑛−1𝑑𝜃=1, for example [26, page 59]. Therefore, |||1𝜔^𝑛𝑓(𝜔)|||≤𝐶ℎ^𝑛𝑛!^𝑠,𝑛∈ℕ₀,𝜔>0,(2.17) which, together with Theorem 2.6(c), implies that there exist 𝐶>0 and ℎ>0 such that sup_𝜔>0|||𝜔+1𝜔^𝛼𝑓(𝜔)|||≤2^𝛼sup_𝜔>01𝜔^𝛼≤𝐶ℎ_𝛼||𝑓(𝜔)||+sup_𝜔>0𝜔^𝛼||𝑓(𝜔)||𝛼!^𝑠,∀𝛼∈ℕ₀.(2.18) By Remark 2.5 there exist 𝐶>0 and ℎ>0 such that sup_𝜔>0𝑒^{ℎ(𝜔+1/𝜔)1/𝑠}||𝑓(𝜔)||≤𝐶(2.19) as claimed.

3. Wavelet Transform

In this section, we define the wavelet transform and study its decay and continuity properties.

The wavelet transform 𝑊(𝑓,𝑔)=𝑊_𝑔𝑓∶𝐿²(ℝ)×𝐿²(ℝ)↦𝐿²(ℝ×ℝ₊) of 𝑓∈𝐿²(ℝ) with respect to the wavelet 𝑔∈𝐿²(ℝ) is given by 𝑊_𝑔𝑓(𝑏,𝑎)=𝑓(𝑥),1𝑎⁻𝑔𝑥−𝑏𝑎=_ℝ𝑓(𝑥)1𝑎⁻𝑔𝑥−𝑏𝑎𝑑𝑥,(𝑏,𝑎)∈ℝ×ℝ⁺,(3.1) and can be extended to any dual pair of spaces provided that duality is meaningful.

In particular, 𝑊_𝑔𝑓 is well defined when 𝑓,𝑔∈𝑆^𝜇𝜈(ℝ). Theorem 2.6 (e) suggests to observe the behavior of 𝑓 and 𝑔 in the configuration and in the frequency domain separately in the study of 𝑊_𝑔𝑓 when 𝑓,𝑔∈𝑆^𝜇𝜈(ℝ).

We introduce following notation for the subexponential weight function: 𝑚_ℎ,𝜈(𝑥)∶=𝑒^{ℎ|𝑥|1/𝜈},𝑥∈ℝ,(3.2) where ℎ,𝜈∈ℝ.

When ℎ>0 and 𝜈>1 the weight 𝑚_ℎ,𝜈 is radial, submultiplicative and subconvolutive. Recall, a nonnegative function 𝜐 is called subconvolutive if 𝜐⁻¹∈𝐿¹(ℝ) and 𝜐⁻¹∗𝜐⁻¹≤𝐶𝜐⁻¹,(3.3) for some constant 𝐶>0.

When 𝜈>1, the following lemma follows from [12, Lemma 7.1], see also [27], where it is proved for a general subconvolutive weight. We give here the proof when the weight is subexponential, since similar arguments will be used later on. Moreover, we allow 𝜈=1 here.

Lemma 3.1. Let there be given 𝜈≥1, ℎ>0 and functions 𝑓 and 𝑔 such that ||𝑓(𝑥)||≤𝐶𝑚^−1ℎ,𝜈(𝑥),||𝑔(𝑥)||≤𝐶𝑚^−1ℎ,𝜈(𝑥),𝑥∈ℝ(3.4) for some positive constant 𝐶. Then, (𝑓∗𝑔)(𝑥)=_ℝ𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑦≤𝐶𝑚^{−1ℎ,𝜈}(𝑥),𝑥∈ℝ,(3.5) for some 𝐶>0 and for every ℎ such that ℎ≤ℎ>0.

Proof. By the elementary inequality |𝑦|^1/𝜈≤|𝑥−𝑦|^1/𝜈+|𝑥|^1/𝜈,𝑥,𝑦∈ℝ,𝜈≥1, it follows that for a given ℎ>0 we have ℎ||𝑥−𝑦||^1/𝜈≥ℎ||𝑦||^1/𝜈−|𝑥|^1/𝜈,𝑥,𝑦∈ℝ,∀ℎsuchthatℎ≥ℎ>0.(3.6) Therefore, 𝑚_ℎ,𝜈(𝑥−𝑦)≥𝑚_ℎ,𝜈(𝑦)𝑚^{−1ℎ,𝜈}(𝑥).(3.7) By the assumption (3.4), it follows that (𝑓∗𝑔)(𝑦)≤𝐶_ℝ𝑚^−1ℎ,𝜈(𝑥)𝑚^−1ℎ,𝜈(𝑥−𝑦)𝑑𝑥≤𝐶_ℝ𝑚^−1ℎ,𝜈(𝑥)𝑚^ℎ,𝜈(𝑥)≤𝐶𝑚_ℎ,𝜈(𝑦)𝑑𝑥_ℝ𝑚^−1ℎ−ℎ_,𝜈(𝑥)𝑚^{−1ℎ,𝜈}(𝑦)𝑑𝑥≤𝐶𝑚^{−1ℎ,𝜈}(𝑦),(3.8) where we choose ℎ such that ℎ>ℎ>0. The proof is complete.

Proposition 3.2. Let there be given 𝜈≥1,ℎ>0 and let 𝑓 and 𝑔 satisfy (3.4) for some 𝐶>0. Then, there exist 𝐶>0 such that ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑚^−1ℎ,𝜈1+𝑏𝑎,(𝑏,𝑎)∈ℝ×ℝ⁺.(3.9)

Proof. When 𝑎=1 the so-called “voice transform” 𝑊_𝑔𝑓(𝑏,1) is just the convolution between 𝑓 and 𝑔, so by Lemma 3.1 we have ||𝑊_𝑔𝑓(𝑏,1)||≤𝐶𝑚^{−1ℎ,𝜈}(𝑏),(3.10) for every ℎ such that ℎ>ℎ>0. In particular, when ℎ=ℎ(1+𝑎)^−1/𝜈=ℎ2^−1/𝜈, we obtain ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑚^−1ℎ,𝜈1+𝑏𝑎,(3.11) for some 𝐶>0. If 𝑎>1, then ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑎_ℝ𝑚^−1ℎ,𝜈(𝑥)𝑚^−1ℎ,𝜈𝑥−𝑏≤𝐶𝑎𝑑𝑥𝑎_ℝ𝑚^−1ℎ,𝜈(𝑥)𝑚^ℎ,𝜈(𝑥/𝑎)𝑚_ℎ,𝜈(𝑏/𝑎)𝑑𝑥=𝐶𝑎𝑚^−1ℎ,𝜈𝑎𝑏_ℝ𝑚_ℎ,𝜈(𝑥/𝑎)≤𝐶𝑚_ℎ,𝜈(𝑥)𝑑𝑥𝑎𝑚^−1ℎ,𝜈1+𝑏𝑎≤𝐶𝑚^−1ℎ,𝜈1+𝑏𝑎,(𝑏,𝑎)∈ℝ×ℝ⁺.(3.12) When 𝑎∈(0,1), by the symmetry property of the wavelet transform: 𝑊_𝑔𝑓(𝑏,𝑎)=1𝑎𝑊^−𝑓−𝑔−𝑏𝑎,𝑎1,(𝑏,𝑎)∈ℝ×ℝ⁺(3.13) (cf. [9, (2.0.1)]) and by the first part of the proof we have: ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑎11/𝑎𝑒1^{ℎ|−𝑏/𝑎/1+1/𝑎|1/𝜈}=𝐶𝑚^−1ℎ,𝜈𝑏1+𝑎,(𝑏,𝑎)∈ℝ×ℝ⁺.(3.14) The proof is complete.

Proposition 3.2 can be compared to [9], where an extra factor (1+𝑎)⁻¹ appears, and where the different technique is used for the proof. Note that (sub)exponential growth of 𝑚_ℎ,𝜈 leads to the convergent integral, _ℝ𝑚_ℎ,𝜈(𝑥/𝑎)𝑚_ℎ,𝜈(𝑥)𝑑𝑥,(3.15) whereas the same type of estimate for a weight of polynomial growth of the form ⟨𝑥⟩^𝑠 gives a divergent integral: _ℝ𝑥𝑎^𝑠⟨𝑥⟩^−𝑠𝑑𝑥.(3.16)

Next, we observe almost exponential decay in the frequency domain and refer to [9, Theorem 12.0.1.] for its polynomial decay counterpart.

Proposition 3.3. Let there be given 𝜇,ℎ>0 and let 𝑓 and 𝑔 satisfy ||𝑓(𝜔)||≤𝐶𝑚^−1ℎ,𝜇𝜔+𝜔1,||𝑔̂(𝜔)||≤𝐶𝑚^−1ℎ,𝜇𝜔+𝜔1,𝜔>0,(3.17) for some positive constant 𝐶>0. Then, there exist ℎ>0 and 𝐶>0 such that ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑚^{−1ℎ,𝜇}𝑎+𝑎1,(𝑏,𝑎)∈ℝ×ℝ⁺.(3.18)

Proof. By the Parseval's formula, it follows that 𝑊_𝑔𝑓(𝑏,𝑎)=_ℝ𝑓(𝜔)𝑒^𝑖𝑏𝜔𝑔̂(𝑎𝜔)𝑑𝜔,(𝑏,𝑎)∈ℝ×ℝ₊.(3.19) The change of variables 𝑎𝜔=𝜔′ gives ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶^∞0𝑚^−1ℎ,𝜇=𝐶𝑎𝜔+𝑎1𝜔𝑚^−1ℎ,𝜇𝜔+𝜔1𝑑𝜔𝑎^∞0𝑚^−1ℎ,𝜇𝜔^+1𝜔^𝑚^−1ℎ,𝜇𝜔𝑎^+𝜔𝑎^𝑑𝜔′.(3.20)
When 𝑎>1, we have ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶^∞0𝑚^−1ℎ,𝜇𝜔^+𝜔1_𝑚^−1ℎ,𝜇𝑎^𝜔^+𝑎^1𝜔^𝑑𝜔′,(3.21) where 1/𝑎=𝑎′. Therefore, it is enough to estimate |𝑊_𝑔𝑓(𝑏,𝑎)| when 0<𝑎≤1.
We split 𝑊_𝑔𝑓 in three parts: ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶¹⁰𝑚^−1ℎ,𝜇+𝐶𝑎𝜔+𝑎1𝜔𝑚^−1ℎ,𝜇𝜔+𝜔1𝑑𝜔_1/𝑎21𝑚^−1ℎ,𝜇𝑎𝜔+1+𝐶𝑎𝜔𝑚^−1ℎ,𝜇𝜔+𝜔1𝑑𝜔_∞1/𝑎2𝑚^−1ℎ,𝜇𝑎𝜔+1𝑎𝜔𝑚^−1ℎ,𝜇𝜔+𝜔1𝑑𝜔=𝐼¹+𝐼₂+𝐼₃.(3.22)
When 0<𝑎≤1 and 𝜔∈(0,1), we have 1/𝑎<𝑎𝜔+1/𝑎𝜔, wherefrom 𝐼₁≤𝐶𝑚^−1ℎ,𝜇1𝑎¹⁰𝑚^−1ℎ,𝜇𝜔+1𝜔𝑑𝜔≤𝐶𝑚^−1ℎ,𝜇𝑎1.(3.23) Since 𝑎+1𝑎≤𝑎2when𝑎∈(0,1],(3.24) it follows that 𝐼₁≤𝐶𝑚^{−1ℎ,𝜇}𝑎+1𝑎,whereℎ=2ℎ^1/𝜇.(3.25)
Next, we make the change of variables: 1/𝜔=𝜔′ in 𝐼₂ and 𝐼₃ and then rename 𝜔′ into 𝜔.
By the mean value theorem, 𝐼₂=𝐶^1𝑎₂𝑚^−1ℎ,𝜇𝜔+1𝜔𝑚^−1ℎ,𝜇𝜔𝑎+𝜔𝑎𝑑𝜔𝜔²=𝐶𝑚^−1ℎ,𝜇𝑎𝜃+𝜃𝑎^1𝑎₂𝑚^−1ℎ,𝜇𝜔+1𝜔𝑑𝜔𝜔²≤𝐶𝑚^−1ℎ,𝜇𝜃𝑎¹⁰𝑚^−1ℎ,𝜇𝜔+1𝜔𝑑𝜔𝜔²≤𝐶𝑚^−1ℎ,𝜇𝜃𝑎≤𝐶𝑚^{−1ℎ,𝜇}𝑎+𝑎1,(3.26) where 𝜃∈(𝑎²,1)⊆(0,1),ℎ=(𝜃/2)^1/𝜇 and we have used inequalities 𝜃/𝑎<𝑎/𝜃+𝜃/𝑎 and (3.24).
It remains to estimate 𝐼₃. We have 𝐼₃=𝐶^𝑎2₀𝑚^−1ℎ,𝜇𝜔+1𝜔𝑚^−1ℎ,𝜇𝜔𝑎+𝜔𝑎𝑑𝜔𝜔²≤𝐶^𝑎2₀𝑚^−1ℎ,𝜇𝜔+1𝜔𝑚^−1ℎ,𝜇𝑎1𝑑𝜔𝜔²≤𝐶𝑚^−1ℎ,𝜇1𝑎¹⁰𝑚^−1ℎ,𝜇𝜔+1𝜔𝑑𝜔𝜔²≤𝐶𝑚^−1ℎ,𝜇1𝑎≤𝐶𝑚^{−1ℎ,𝜇}𝑎+𝑎1,(3.27) where ℎ=ℎ/2^1/𝜇 and we have used inequalities 1/𝑎<𝑎/𝜔+𝜔/𝑎 and (3.24).
This completes the proof.

We end the section with a continuity result when 𝑓,𝑔∈𝑆^𝜈𝜇,+(ℝ). We refer to [9, Theorem 19.0.1] for the case of progressive functions of polynomial decay, and to [19] for the (Gelfand-Shilov) spaces of type 𝑆. First, we introduce the space 𝑆^𝜇𝜈(ℝ×ℝ₊).

Definition 3.4. Let there be given positive constants ℎ,𝜈 and 𝜇. The space 𝑆^𝜇𝜈,ℎ(ℝ×ℝ₊) consists of all 𝜏∈𝐶^∞(ℝ×ℝ₊) such that ||𝜏(𝑏,𝑎)||≤𝐶𝑚^−1ℎ,𝜈1+𝑏𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1,(𝑏,𝑎)∈ℝ×ℝ⁺,(3.28) for some 𝐶>0.
Then 𝑆^𝜇𝜈(ℝ×ℝ₊)=⋃_ℎ>0𝑆^𝜇𝜈,ℎ(ℝ×ℝ₊), and the topology is defined by the inductive limit: 𝑆^𝜇𝜈ℝ×ℝ₊=indlim_ℎ>0𝑆^𝜇𝜈,ℎℝ×ℝ₊.(3.29)

Theorem 3.5. Let there be given 𝜈≥1 and 𝜇>0 and let 𝑓,𝑔∈𝑆^𝜈𝜇,+(ℝ). Then, the wavelet transform 𝑊∶(𝑓,𝑔)↦𝑊_𝑔𝑓 is a continuous map from 𝑆^𝜈𝜇,+(ℝ)×𝑆^𝜈𝜇,+(ℝ) into 𝑆^𝑠𝜈(ℝ×ℝ₊), where 𝑠=max{𝜈,𝜇}.

Proof. Since 𝑓 and 𝑔 are smooth, 𝑊_𝑔𝑓(𝑏,𝑎)∈𝐶^∞(ℝ×ℝ₊). Proposition 2.8 implies that ||𝑓(𝜔)||≤𝐶𝑚^−1ℎ,𝑠𝜔+𝜔1,||𝑔̂(𝜔)||≤𝐶𝑚^−1ℎ,𝑠𝜔+𝜔1,𝜔>0,(3.30) for 𝑠=max{𝜈,𝜇} and for some positive constants ℎ and 𝐶.
By Propositions 3.2 and 3.3, it follows that there exist positive constants 𝐶 and ℎ such that ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑚^{−1ℎ/2,𝜈}1+𝑏𝑎𝑚^{−1ℎ/2,𝑠}𝑎+𝑎1,(𝑏,𝑎)∈ℝ×ℝ⁺,(3.31) which proves the continuity.

Note that 𝑓 and 𝑔 in Theorem 3.5 belong to the same progressive Gelfand-Shilov space, while in [19] different decay conditions are imposed on 𝑓 and 𝑔. Since the analyzing wavelet 𝑔 in [19] may not satisfy the admissibility condition (4.24), the results given there are of global nature, and the inversion formula cannot be derived.

4. Inverse Wavelet Transform

In this section, we study the inverse wavelet transform, or the wavelet synthesis operator 𝑀_𝑔 of function 𝜏∈𝑆^𝜇𝜈(ℝ×ℝ₊) with respect to the wavelet 𝑔∈𝐿¹(ℝ) given by 𝑀_𝑔𝜏(𝑥)=^0+∞𝑑𝑎𝑎_ℝ𝜏(𝑏,𝑎)1𝑎𝑔𝑥−𝑎𝑏𝑑𝑏,𝑥∈ℝ.(4.1)

Our first result is analogous to Proposition 3.2 (see also [9, Theorem 18.2.1]).

Proposition 4.1. Let 𝜈≥1 and 𝜇>0. If 𝜏∈𝑆^𝜇𝜈(ℝ×ℝ₊) and if 𝑔 satisfies ||𝑔(𝑥)||≤𝐶𝑚^−1ℎ,𝜈(𝑥),𝑥∈ℝ,(4.2) for some positive constants ℎ and 𝐶, then ||𝑀_𝑔𝜏(𝑥)||≤𝐶𝑚^{−1ℎ,𝜈}(𝑥),𝑥∈ℝ,(4.3) for some positive constants ℎ and 𝐶.

Proof. We have ||𝑀_𝑔𝜏(𝑥)||≤^0+∞𝑑𝑎𝑎_ℝ||𝜏(𝑏,𝑎)||1≤𝐶_0+∞𝑎|||𝑔𝑥−𝑎𝑏|||𝑑𝑏𝑑𝑎𝑎²_ℝ𝑚^−1ℎ,𝜈𝑏≤𝐶𝐼₁1+𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑚^−1ℎ,𝜈𝑥−𝑎𝑏𝑑𝑏(𝑥)+𝐼₂(𝑥),𝑥∈ℝ,(4.4) where in 𝐼₁ the integration with respect to 𝑎 is taken over (0,1) and in 𝐼₂ over (1,∞).
By (3.6), it follows that for any ℎ∈(0,ℎ) we have 𝐼₁(𝑥)≤¹⁰𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎𝑎²_ℝ𝑚^−1ℎ,𝜈𝑏1+𝑎𝑚^{−1ℎ,𝜈}(𝑥)𝑚_ℎ,𝜈(𝑏)𝑑𝑏,𝑥∈ℝ.(4.5) Since _ℝ𝑚^−1ℎ,𝜈𝑏1+𝑎𝑚^ℎ,𝜈(𝑏)𝑑𝑏<∞,(4.6) when ℎ<ℎ2^−1/𝜈≤ℎ(1+𝑎)^−1/𝜈 and since ¹⁰𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎𝑎²<∞,(4.7) it follows that 𝐼₁(𝑥)≤𝐶𝑚^{−1ℎ,𝜈}(𝑥),𝑥∈ℝ,(4.8) for some 𝐶>0.
Next, we estimate 𝐼₂. We have 𝐼₂(𝑥)=^1+∞𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎_ℝ1≤𝑎𝑚^−1ℎ,𝜈1+𝑏𝑎𝑎1𝑚^−1ℎ,𝜈𝑥−𝑎𝑏𝑑𝑏_1+∞𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎_ℝ1=𝑎𝑚^−1ℎ,𝜈2𝑏𝑎𝑎1𝑚^−1ℎ,𝜈𝑥2−𝑎𝑏𝑑𝑏_1+∞𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎_ℝ𝐷_𝑎𝑚^−1ℎ,𝜈𝑏=2𝐷^𝑎𝑚^−1ℎ,𝜈𝑥−2𝑏𝑑𝑏_1+∞𝑚^−1ℎ,𝜇𝑎+1=𝑎𝐷^𝑎𝑚^−1ℎ,𝜈2𝑏∗𝐷^𝑎𝑚^−1ℎ,𝜈2𝑏(𝑥)𝑑𝑎_1+∞𝑚^−1ℎ,𝜇𝑎+1𝑎𝐷^𝑎𝑚^−1ℎ,𝜈2𝑏∗𝑚^−1ℎ,𝜈2𝑏(𝑥)𝑑𝑎,𝑥∈ℝ.(4.9)
By Lemma 3.1, 𝐼₂(𝑥)≤𝐶^1+∞𝑚^−1ℎ,𝜇𝑎+1𝑎𝐷^𝑎𝑚^{−1ℎ,𝜈}(𝑥)𝑑𝑎,𝑥∈ℝ,(4.10) for every ℎ∈(0,ℎ). After the change of variables 𝑎′=1/𝑎 and renaming 𝑎′ to 𝑎, by the mean value theorem, we obtain 𝐼₂(𝑥)≤¹⁰𝐶𝑚^−1ℎ,𝜇𝑎+1𝑎𝑚^{−1ℎ,𝜈}(𝑎𝑥)𝑑𝑎=𝐶𝑚_{−1ℎ,𝜈}𝑎(𝜃𝑥)¹⁰𝑚^−1ℎ,𝜇𝑎+1𝑎𝑑𝑎𝑎,𝑥∈ℝ,(4.11) for some 𝜃∈(0,1). Therefore, there are 𝐶>0 and ℎ>0 such that 𝐼₂(𝑥)≤𝐶𝑚^{−1ℎ,𝜈}(𝑥),𝑥∈ℝ, and (4.3) is proved.

The next result can be compared to [9, Theorem 18.1.1]. Here, the assumption on the decay of 𝑔 leads to a more precise estimate for 𝑀_𝑔𝜏 then the one given in [9].

Proposition 4.2. Let 𝜈 and 𝜇 be positive constants. If 𝜏∈𝑆^𝜇𝜈(ℝ×ℝ₊) and if 𝑔 satisfies ||𝑔̂(𝜔)||≤𝐶𝑚^−1ℎ,𝜇𝜔+𝜔1,𝜔>0,(4.12) for some positive constants ℎ and 𝐶, then |||𝑀_𝑔𝜏(𝜔)|||≤𝐶𝑚^{−1ℎ,𝜇}𝜔+1𝜔,𝜔>0,(4.13) for some positive constants ℎ and 𝐶.

Proof. The Fourier transform of 𝑀_𝑔𝜏 is given by 𝑀_𝑔𝜏(𝜔)=^0+∞𝑑𝑎𝑎_ℝ𝜏(𝑏,𝑎)𝑔̂(𝑎𝜔)𝑒^{−𝑖𝑏𝜔}𝑑𝑏,𝜔>0.(4.14) Therefore, |||𝑀_𝑔𝜏(𝜔)|||≤𝐶^0+∞1𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑚^−1ℎ,𝜇𝑎𝜔+𝑎1𝜔𝑑𝑎_ℝ𝑚^−1ℎ,𝜈𝑏≤𝐶₁1+𝑎𝑑𝑏𝐼₁(𝜔)+𝐼₂(𝜔),𝜔>0,(4.15) where 𝐼₁(𝜔)=¹⁰1+𝑎𝐼𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑚^−1ℎ,𝜇𝑎𝜔+𝑎1𝜔𝑑𝑎,₂(𝜔)=^1+∞1+𝑎𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑚^−1ℎ,𝜇𝑎𝜔+𝑎1𝜔𝑑𝑎,𝜔>0,(4.16) and we have made the change of variables 𝑏/(1+𝑎)=𝑏′.
We use similar arguments as in proof of Proposition 3.3 to estimate 𝐼₁(𝜔) when 0<𝜔≤1. This gives 𝐼₁(𝜔)≤𝑚^−1ℎ,𝜇1𝜔¹⁰2𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑑𝑎≤𝑚^−1ℎ,𝜇21𝜔+𝜔1,𝜔∈(0,1],(4.17) where we have used the inequality (𝜔+1/𝜔)/2<1/𝜔,𝜔∈(0,1].
When 𝜔>1 by the mean value theorem, we obtain 𝐼₁(𝜔)≤𝐶𝑚^−1ℎ,𝜇𝜃𝜔+1𝜃𝜔¹⁰2≤𝐶𝑚_−1ℎ,𝜇𝑎𝑚^−1ℎ,𝜇𝑎+𝑎1𝑑𝑎(𝜃𝜔)≤𝐶𝑚^{−1ℎ,𝜇}𝜔+1𝜔,𝜔∈(1,∞),(4.18) where 𝜃∈(0,1) and ℎ=(𝜃/2)^1/𝜇ℎ.
It remains to estimate 𝐼₂(𝜔),𝜔>0.
When 𝜔∈(0,1), by change of variables 1/𝑎=𝑎′ and by the mean value theorem, it follows that 𝐼₂(𝜔)=¹⁰1+𝑎^(𝑎^)²𝑚^−1ℎ,𝜇𝑎^+1𝑎^𝑚^−1ℎ,𝜇𝑎^𝜔+𝑎^≤𝜔𝑑𝑎′₁₀2(𝑎^)²𝑚^−1ℎ,𝜇𝑎^+1=𝑚_{−1ℎ𝜃1/𝜇,𝜇}𝑎^𝑚^−1ℎ,𝜇𝑎^𝜔+𝜔1𝑑𝑎′𝜔+𝜔1¹⁰2(𝑎^)²𝑚^−1ℎ,𝜇𝑎^+1≤𝐶𝑚_{−1ℎ,𝜇}𝑎^𝑑𝑎′𝜔+𝜔1,𝜔∈(0,1),(4.19) where 𝜃∈(0,1) and ℎ=𝜃^1/𝜇ℎ.
In a similar way, as in the proof of Proposition 3.3, it follows that 𝐼₂(𝜔)≤𝐶𝑚^−1ℎ2_{−1/𝜇,𝜇}𝜔+1𝜔,𝜔∈(1,∞),(4.20) and the proof is complete.

Finally, we have the following result, which can be compared to the inversion formulas from [14, 15] which hold for general coorbit spaces. As mentioned in the introduction, our approach is different due to the fact that the weight 𝑚_ℎ,𝜈 is not submultiplicative on the “𝑎𝑥+𝑏”-group.

Theorem 4.3. Let 𝜈≥1 and 𝜇>0. The inverse wavelet transform 𝑀∶(𝑔,𝜏)↦𝑀_𝑔𝜏 is a continuous map from 𝑆^𝜈𝜇,+(ℝ)×𝑆^𝜇𝜈(ℝ×ℝ₊) into 𝑆^𝜈𝑠,+(ℝ), where 𝑠=max{𝜈,𝜇}.
Moreover, if 𝑔≠0, then for any 𝑓∈𝑆^𝜈𝑠,+(ℝ) the following inversion formula holds: 𝑓(𝑥)=1𝑐_𝑔^∞0𝑑𝑎𝑎_ℝ𝑊_𝑔𝑓(𝑏,𝑎)1𝑎𝑔𝑥−𝑎𝑏𝑑𝑏,(4.21) where 𝑐_𝑔=∫^∞0|ĝ(𝜔)|²(𝑑𝜔/𝜔).

Proof. For the first part of Theorem 4.3, it is enough to observe that Proposition 2.8 implies that ||𝑔̂(𝜔)||≤𝐶𝑚^−1ℎ,𝑠𝜔+𝜔1,𝜔>0,(4.22) for some positive constants ℎ and 𝐶, and for 𝑠=max{𝜈,𝜇}.
The continuity now follows from Propositions 4.1 and 4.2 and Theorem 2.6(e), since |||𝑀_𝑔𝜏(𝜔)|||≤𝐶𝑚^{−1ℎ,𝑠}𝜔+1𝜔≤𝐶𝑚^{−1ℎ,𝑠}(𝜔),𝜔>0.(4.23)
From the the proof of Proposition 2.8, it follows that 𝑔 satisfies the admissibility condition: 0<||||^∞0||𝑔̂(𝜔)||²𝑑𝜔𝜔||||<∞.(4.24) Since 1𝑐_𝑔^∞0𝑑𝑎𝑎_ℝ𝑊_𝑔𝑓(𝑏,𝑎)1𝑎𝑔𝑥−𝑎𝑏𝑑𝑏=𝑐1_𝑔𝑀_𝑔𝑊_𝑔𝑓(𝑥),𝑥∈ℝ,(4.25) and by Theorem 3.5, we have that 𝑊_𝑔𝑓∈𝑆^𝑠𝜈(ℝ×ℝ₊), from the first part of the proof it follows that 𝑀_𝑔(𝑊_𝑔𝑓)∈𝑆^𝜈𝑠,+(ℝ).
The proof is now the same as the proof of [9, Theorem 14.0.2] if we take 𝑔 to be its own reconstruction wavelet and observe 𝑆^𝜈𝑠,+(ℝ) instead of the space of polynomially strip localized functions.

5. Concluding Remarks

We first note that under the assumptions of Proposition 3.2 we have that for any given 𝑀>0 there exist ℎ>0 and 𝐶>0 such that (i)|𝑊_𝑔𝑓(𝑏,𝑎)|≤𝐶𝑒⁻ℎ_|𝑏|^1/𝜈,(𝑏,𝑎)∈ℝ×(0,𝑀]; (ii)|𝑊_𝑔𝑓(𝑏,𝑎)|≤𝐶𝐷_𝑎(𝑒⁻ℎ_|⋅|^1/𝜈)(𝑏),(𝑏,𝑎)∈ℝ×[𝑀,∞).

Note that those estimates imply that we are not able to separate variables 𝑏 and 𝑎 in a straightforward manner and lead to (3.28) in Definition 3.4. It is probably more natural to have a possibility to analyze decay with respect to variables 𝑏 and 𝑎 separately. Therefore, we have the following definition.

Definition 5.1. Let there be given positive constants ℎ,𝜈, and 𝜇. The space 𝑍^𝜇𝜈,ℎ(ℝ×ℝ₊) consists of all 𝜏∈𝐶^∞(ℝ×ℝ₊) such that ||𝜏(𝑏,𝑎)||≤𝐶𝑚^−1ℎ,𝜈(𝑏)𝑚^−1ℎ,𝜇𝑎+𝑎1,(𝑏,𝑎)∈ℝ×ℝ⁺,(5.1) for some 𝐶>0.
Then 𝑍^𝜇𝜈(ℝ×ℝ₊)=⋃_ℎ>0𝑍^𝜇𝜈,ℎ(ℝ×ℝ₊), and the topology is given by the inductive limit: 𝑍^𝜇𝜈ℝ×ℝ₊=indlim_ℎ>0𝑍^𝜇𝜈,ℎℝ×ℝ₊.(5.2)
Next, we reformulate results from Sections 3 and 4 in terms of 𝑍^𝜇𝜈(ℝ×ℝ₊) since we believe that this might be of independent interest.

Theorem 5.2. Let 𝜈>1 and 𝜇>0, let 𝑓,𝑔∈𝑆^{𝜇,+𝜈−𝜀}(ℝ), with 𝜀>0 chosen such that 𝜈−𝜀≥1. Then, the wavelet transform 𝑊∶(𝑓,𝑔)↦𝑊_𝑔𝑓 is a continuous map from 𝑆^{𝜇,+𝜈−𝜀}(ℝ)×𝑆^{𝜇,+𝜈−𝜀}(ℝ) into 𝑍^𝑠𝜈(ℝ×ℝ₊), where 𝑠=max{𝜈,𝜇}.

Proof. It follows from the proof of Theorem 3.5 and the fact that for every positive 𝑁,𝑀, and 𝜀 there is a constant 𝐶>0 such that 𝑒^{−𝑁|⋅|1/(𝜈−𝜀)}≤𝐶𝑒^{−𝑀|⋅|1/𝜈},(5.3) wherefrom ||𝑊_𝑔𝑓(𝑏,𝑎)||≤𝐶𝑒ℎ^{|𝑏/(1+𝑎)|1/(𝜈−𝜀)}≤𝐶𝑒^{𝑘|𝑏|1/𝜈},(𝑏,𝑎)∈ℝ×ℝ₊,(5.4) for arbitrary 𝑘>0. The theorem is proved.

The inverse wavelet transform 𝑀_𝑔 of 𝜏∈𝑍^𝜇𝜈(ℝ×ℝ₊) with respect to the wavelet 𝑔∈𝐿¹(ℝ) given by 𝑀_𝑔𝜏(𝑥)=^0+∞𝑑𝑎𝑎_ℝ𝜏(𝑏,𝑎)1𝑎𝑔𝑥−𝑎𝑏𝑑𝑏,𝑥∈ℝ.(5.5)

Propositions 4.1 and 4.2 still hold true with 𝑆^𝜇𝜈(ℝ×ℝ₊) replaced by 𝑍^𝜇𝜈(ℝ×ℝ₊), while Theorem 4.3 takes the following form.

Theorem 5.3. Let 𝜈≥1 and 𝜇>0. The inverse wavelet transform 𝑀∶(𝑔,𝜏)↦𝑀_𝑔𝜏 is a continuous map from 𝑆^𝜈𝜇,+(ℝ)×𝑍^𝜇𝜈(ℝ×ℝ₊) into 𝑆^𝜈𝑠,+(ℝ), where 𝑠=max{𝜈,𝜇}.
Moreover, if 𝑓,𝑔∈𝑆^{𝜇,+𝜈−𝜀}(ℝ) for some 𝜀>0 such that 𝜈−𝜀>1 and 𝑔≠0, then 𝑓(𝑥)=1𝑐_𝑔^∞0𝑑𝑎𝑎_ℝ𝑊_𝑔𝑓(𝑏,𝑎)1𝑎𝑔𝑥−𝑎𝑏𝑑𝑏,(5.6) where 𝑐_𝑔=∫^∞0|𝑔̂(𝜔)|²(𝑑𝜔/𝜔).

Acknowledgment

The research was supported by MPN of Serbia, Project nos. 174024, III 44006 and by PSNTR Project no. 114-451-2167.