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 [14] 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() which consists of functions in 𝐿2() whose Fourier transforms are supported by the right half-plane. It is well known that Hardy spaces 𝐻𝑝(),1𝑝, play fundamental role in signal processing, cf. [57]. Analytic signal 𝐹, associated to a real-valued signal 𝑓𝐿2() by 𝐹=𝑓+1𝐻(𝑓), where 𝐻(𝑓) is the Hilbert transform of 𝑓, belongs to 𝐻2(), 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, 79].

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 [911]. 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 𝑒𝑎||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 𝑤𝑠(𝑥,𝜉)=𝑤𝑠(𝜉)=𝜉𝑠,𝑠>0, and a weight of the form 𝑤𝑠1,𝑠2(𝑥,𝜉)=𝑥𝑠1𝜉𝑠2, 𝑠1,𝑠2>0, (𝑥,𝜉)×(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 [1315]. This is not the case for 𝑤𝑠1,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. [1618]). 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, (𝐿𝑝(),𝑝),1𝑝, are Lebesgue spaces, 𝑓,𝑔 denotes the dual pairing of 𝑓 and 𝑔, and the Fourier transform of 𝑓𝐿1() is normalized to be 𝑓(𝜉)=𝑓(𝜉)=𝑒2𝜋𝑖𝑥𝜉𝑓(𝑥)𝑑𝑥,𝜉.(1.1) By 𝐷𝑎 we denote the dilation 𝐷𝑎()=(1/𝑎)𝑓(/𝑎),𝑎+=(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 𝑆𝜈𝜇,+() are introduced in Definition 2.1. The range of the wavelet transform over such spaces is contained in 𝑆𝜇𝜈(×+), see Definition 3.4, and in 𝑍𝜇𝜈(×+), 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 ,𝜈, and 𝜇 be positive constants such that 𝜈+𝜇1. By 𝑆𝜇𝜈,() we denote the Banach space which consists of all 𝑓𝐶() such that the norm 𝑓𝑆𝜇𝜈,=sup𝛼,𝛽0𝑥𝛼𝑓(𝛽)(𝑥)𝐿𝛼+𝛽𝛼!𝜈𝛽!𝜇(2.1) is finite.

The Gelfand-Shilov space 𝑆𝜇𝜈() is given by 𝑆𝜇𝜈()=>0𝑆𝜇𝜈,(),(2.2) and its topology is given by the inductive limit: 𝑆𝜇𝜈()=indlim>0𝑆𝜇𝜈,().(2.3)

Definition 2.1. Let ,𝜈, and 𝜇, be positive constants such that 𝜈+𝜇1. The space 𝑆𝜇,+𝜈,() consists of all 𝑓𝐶() such that supp𝑓[0,) and such that (2.1) is finite.
The progressive Gelfand-Shilov space 𝑆𝜈𝜇,+() is given by 𝑆𝜈𝜇,+()=>0𝑆𝜇,+𝜈,(),(2.4) and its topology is given by the inductive limit: 𝑆𝜈𝜇,+()=indlim>0𝑆𝜇,+𝜈,().(2.5)

Obviously, 𝑆𝜈𝜇,+() is a closed subspace of 𝑆𝜇𝜈().

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 𝑆𝜇,+𝜈,(). For the projective limits of 𝑆𝜇𝜈,(), >0, we refer to [21, 22].
Note also that the decay of 𝑥𝛼𝑓(𝛽)(𝑥)𝐿 can be controlled by some other choice of sequences {𝑀𝛼},{𝑁𝛽}, instead of the Gevrey sequences 𝑀𝛼=𝛼!𝜈 and 𝑁𝛽=𝛽!𝜇 used in (2.1). We refer to [1, 2123] for the usual conditions on {𝑀𝛼} and {𝑁𝛽}.
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 𝑆𝜈𝜇,+() and that the same is true for translation and dilation by a positive factor, for example [2, Chapter IV]. On the other hand, 𝑆𝜈𝜇,+() 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 𝑆𝜈𝜇,+(). 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 𝑓𝐶(). Then, the following conditions are equivalent. (a)There exists >0 such that sup𝑥||𝑥𝛼||𝑓(𝑥)𝐶𝛼𝛼!𝑠,𝛼0,(2.6) for some 𝐶>0. (b)There exists 𝑘>0 such that sup𝑥𝑒𝑘|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 𝐶1𝑒(𝑡𝜀)𝜂1/𝑡𝑗=0𝜂𝑗(𝑗!)𝑡𝐶𝑒(𝑡+𝜀)𝜂1/𝑡,𝜂0.(2.8)

Remark 2.5. When 𝑠>0 and 𝑓𝐶() by Proposition 2.3, it directly follows that the following conditions are equivalent. (a)There exists >0 such that sup𝜔>0|||1𝜔+𝜔𝛼|||𝑓(𝜔)𝐶𝛼𝛼!𝑠,𝛼0,(2.9) for some 𝐶>0.(b)There exists 𝑘>0 such that sup𝜔>0𝑒𝑘(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+𝜇1 and let 𝑓𝐶(). Then, the following conditions are equivalent,(a)𝑓𝑆𝜇𝜈(). (b)There exist >0 and 𝐶>0 (which depend only on 𝑓) such that sup𝑥||𝑥𝛼||𝑓(𝑥)𝐶𝛼𝛼!𝜈,𝛼0sup𝑥||𝑓(𝛽)||(𝑥)𝐶𝛽𝛽!𝜇,𝛽0.(2.11)(c)There exist >0 and 𝐶>0 (which depend only on 𝑓) such that sup𝑥||𝑥𝛼𝑓||(𝑥)𝐶𝛼𝛼!𝜈,𝛼0sup𝜔||𝜔𝛽𝑓||(𝜔)𝐶𝛽𝛽!𝜇,𝛽0.(2.12)(d)There exist >0 and 𝐶>0 (which depend only on 𝑓) such that sup𝜔|||𝜔𝛼𝑓(𝛽)|||(𝜔)𝐶𝛼+𝛽𝛼!𝜇𝛽!𝜈,𝛽0.(2.13)(e)There exist >0 and 𝐶>0 (which depend only on 𝑓) such that sup𝑥𝑒|𝑥|1/𝜈||||𝑓(𝑥)𝐶,sup𝜔𝑒|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 𝑆𝜇𝜈()(𝑆𝜇𝜈()=𝑆𝜈𝜇()) 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 𝑆𝜈𝜇,+() where the supremum with respect to 𝜔 in Theorem 2.6(c), (d), and (e) is taken over 𝜔>0 instead of 𝜔.

Moreover, elements in 𝑆𝜈𝜇,+() 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 𝑆𝜈𝜇,+() are almost exponentially strip localized.

Proposition 2.8. Let 𝜈 and 𝜇 be positive constants such that 𝜈+𝜇1 and let 𝑓𝑆𝜇𝜈() satisfy supp𝑓[0,), that is, 𝑓𝑆𝜈𝜇,+(). Then, sup𝜔>0𝑒(𝜔+1/𝜔)1/𝑠|𝑓(𝜔)|<, where 𝑠=max{𝜇,𝜈}.

Proof. Note that 𝑓𝑆𝜈𝜇,+() implies 𝑓(𝑛)(0)=0,𝑛0,  that is, 𝑥𝑛𝑓(𝑥)𝑑𝑥=0,𝑛0,(2.15) and 𝑓(𝜔)=𝑜(𝜔𝑛) when 𝜔0, for every 𝑛0. By Taylor's formula, it follows that ||||=|||||𝑓(𝜔)𝑛1𝑚=0𝑓(𝑚)(0)𝜔𝑚!𝑚+𝜔𝑛𝑛𝑛!10(1𝜃)𝑛1𝑓(𝑛)|||||𝜔(𝜃𝜔)𝑑𝜃𝑛𝑛!sup𝜔>0|||𝑓(𝑛)|||𝑛(𝜔)10(1𝜃)𝑛1𝑑𝜃𝜔𝑛𝐶𝑛𝑛!𝑠,(2.16) where we have used Theorem 2.6(d) and 𝑛10(1𝜃)𝑛1𝑑𝜃=1, for example [26, page 59]. Therefore, |||1𝜔𝑛|||𝑓(𝜔)𝐶𝑛𝑛!𝑠,𝑛0,𝜔>0,(2.17) which, together with Theorem 2.6(c), implies that there exist 𝐶>0 and >0 such that sup𝜔>0|||1𝜔+𝜔𝛼|||𝑓(𝜔)2𝛼sup𝜔>01𝜔𝛼||||𝑓(𝜔)+sup𝜔>0𝜔𝛼||||𝑓(𝜔)𝐶𝛼𝛼!𝑠,𝛼0.(2.18) By Remark 2.5 there exist 𝐶>0 and >0 such that sup𝜔>0𝑒(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)=𝑊𝑔𝑓𝐿2()×𝐿2()𝐿2(×+) of 𝑓𝐿2() with respect to the wavelet 𝑔𝐿2() is given by 𝑊𝑔1𝑓(𝑏,𝑎)=𝑓(𝑥),𝑎𝑔𝑥𝑏𝑎=1𝑓(𝑥)𝑎𝑔𝑥𝑏𝑎𝑑𝑥,(𝑏,𝑎)×+,(3.1) and can be extended to any dual pair of spaces provided that duality is meaningful.

In particular, 𝑊𝑔𝑓 is well defined when 𝑓,𝑔𝑆𝜇𝜈(). 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 𝑊𝑔𝑓 when 𝑓,𝑔𝑆𝜇𝜈().

We introduce following notation for the subexponential weight function: 𝑚,𝜈(𝑥)=𝑒|𝑥|1/𝜈,𝑥,(3.2) where ,𝜈.

When >0 and 𝜈>1 the weight 𝑚,𝜈 is radial, submultiplicative and subconvolutive. Recall, a nonnegative function 𝜐 is called subconvolutive if 𝜐1𝐿1() and 𝜐1𝜐1𝐶𝜐1,(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 𝜈1, >0 and functions 𝑓 and 𝑔 such that ||||𝑓(𝑥)𝐶𝑚1,𝜈||||(𝑥),𝑔(𝑥)𝐶𝑚1,𝜈(𝑥),𝑥(3.4) for some positive constant 𝐶. Then, (𝑓𝑔)(𝑥)=𝑓(𝑦)𝑔(𝑥𝑦)𝑑𝑦𝐶𝑚1,𝜈(𝑥),𝑥,(3.5) for some 𝐶>0 and for every such that >0.

Proof. By the elementary inequality |𝑦|1/𝜈|𝑥𝑦|1/𝜈+|𝑥|1/𝜈,𝑥,𝑦,𝜈1, it follows that for a given >0 we have ||||𝑥𝑦1/𝜈||𝑦||1/𝜈|𝑥|1/𝜈,𝑥,𝑦,suchthat>0.(3.6) Therefore, 𝑚,𝜈(𝑥𝑦)𝑚,𝜈(𝑦)𝑚1,𝜈(𝑥).(3.7) By the assumption (3.4), it follows that 𝐶(𝑓𝑔)(𝑦)𝑚1,𝜈(𝑥)𝑚1,𝜈𝐶(𝑥𝑦)𝑑𝑥𝑚1,𝜈𝑚(𝑥),𝜈(𝑥)𝑚,𝜈(𝐶𝑦)𝑑𝑥𝑚1,𝜈(𝑥)𝑚1,𝜈(𝑦)𝑑𝑥𝐶𝑚1,𝜈(𝑦),(3.8) where we choose such that >>0. The proof is complete.

Proposition 3.2. Let there be given 𝜈1,>0 and let 𝑓 and 𝑔 satisfy (3.4) for some 𝐶>0. Then, there exist 𝐶>0 such that ||𝑊𝑔||𝑓(𝑏,𝑎)𝐶𝑚1,𝜈𝑏1+𝑎,(𝑏,𝑎)×+.(3.9)

Proof. When 𝑎=1 the so-called “voice transform” 𝑊𝑔𝑓(𝑏,1) is just the convolution between 𝑓 and 𝑔, so by Lemma 3.1 we have ||𝑊𝑔||𝑓(𝑏,1)𝐶𝑚1,𝜈(𝑏),(3.10) for every such that >>0. In particular, when =(1+𝑎)1/𝜈=21/𝜈, we obtain ||𝑊𝑔||𝑓(𝑏,𝑎)𝐶𝑚1,𝜈𝑏,1+𝑎(3.11) for some 𝐶>0. If 𝑎>1, then ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)𝑎𝑚1,𝜈(𝑥)𝑚1,𝜈𝑥𝑏𝑎𝐶𝑑𝑥𝑎𝑚1,𝜈𝑚(𝑥),𝜈(𝑥/𝑎)𝑚,𝜈𝐶(𝑏/𝑎)𝑑𝑥=𝑎𝑚1,𝜈𝑏𝑎𝑚,𝜈(𝑥/𝑎)𝑚,𝜈𝐶(𝑥)𝑑𝑥𝑎𝑚1,𝜈𝑏1+𝑎𝐶𝑚1,𝜈𝑏1+𝑎,(𝑏,𝑎)×+.(3.12) When 𝑎(0,1), by the symmetry property of the wavelet transform: 𝑊𝑔1𝑓(𝑏,𝑎)=𝑎𝑊𝑓𝑔𝑏𝑎,1𝑎,(𝑏,𝑎)×+(3.13) (cf. [9, (2.0.1)]) and by the first part of the proof we have: ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)𝑎111/𝑎𝑒|𝑏/𝑎/1+1/𝑎|1/𝜈=𝐶𝑚1,𝜈𝑏1+𝑎,(𝑏,𝑎)×+.(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,𝜈 leads to the convergent integral, 𝑚,𝜈(𝑥/𝑎)𝑚,𝜈(𝑥)𝑑𝑥,(3.15) whereas the same type of estimate for a weight of polynomial growth of the form 𝑥𝑠 gives a divergent integral: 𝑥𝑎𝑠𝑥𝑠𝑑𝑥.(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,>0 and let 𝑓 and 𝑔 satisfy ||||𝑓(𝜔)𝐶𝑚1,𝜇1𝜔+𝜔,||||̂𝑔(𝜔)𝐶𝑚1,𝜇1𝜔+𝜔,𝜔>0,(3.17) for some positive constant 𝐶>0. Then, there exist >0 and 𝐶>0 such that ||𝑊𝑔||𝑓(𝑏,𝑎)𝐶𝑚1,𝜇1𝑎+𝑎,(𝑏,𝑎)×+.(3.18)

Proof. By the Parseval's formula, it follows that 𝑊𝑔𝑓(𝑏,𝑎)=𝑓(𝜔)𝑒𝑖𝑏𝜔̂𝑔(𝑎𝜔)𝑑𝜔,(𝑏,𝑎)×+.(3.19) The change of variables 𝑎𝜔=𝜔 gives ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)0𝑚1,𝜇1𝑎𝜔+𝑚𝑎𝜔1,𝜇1𝜔+𝜔=𝐶𝑑𝜔𝑎0𝑚1,𝜇𝜔+1𝜔𝑚1,𝜇𝜔𝑎+𝑎𝜔𝑑𝜔.(3.20)
When 𝑎>1, we have ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)0𝑚1,𝜇𝜔+1𝜔𝑚1,𝜇𝑎𝜔+1𝑎𝜔𝑑𝜔,(3.21) where 1/𝑎=𝑎. Therefore, it is enough to estimate |𝑊𝑔𝑓(𝑏,𝑎)| when 0<𝑎1.
We split 𝑊𝑔𝑓 in three parts: ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)10𝑚1,𝜇1𝑎𝜔+𝑚𝑎𝜔1,𝜇1𝜔+𝜔+𝐶𝑑𝜔1/𝑎21𝑚1,𝜇1𝑎𝜔+𝑚𝑎𝜔1,𝜇1𝜔+𝜔+𝐶𝑑𝜔1/𝑎2𝑚1,𝜇1𝑎𝜔+𝑚𝑎𝜔1,𝜇1𝜔+𝜔𝑑𝜔=𝐼1+𝐼2+𝐼3.(3.22)
When 0<𝑎1 and 𝜔(0,1), we have 1/𝑎<𝑎𝜔+1/𝑎𝜔, wherefrom 𝐼1𝐶𝑚1,𝜇1𝑎10𝑚1,𝜇1𝜔+𝜔𝑑𝜔𝐶𝑚1,𝜇1𝑎.(3.23) Since 1𝑎+𝑎2𝑎],when𝑎(0,1(3.24) it follows that 𝐼1𝐶𝑚1,𝜇1𝑎+𝑎,where=21/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=𝐶1𝑎2𝑚1,𝜇1𝜔+𝜔𝑚1,𝜇𝑎𝜔+𝜔𝑎𝑑𝜔𝜔2=𝐶𝑚1,𝜇𝑎𝜃+𝜃𝑎1𝑎2𝑚1,𝜇1𝜔+𝜔𝑑𝜔𝜔2𝐶𝑚1,𝜇𝜃𝑎10𝑚1,𝜇1𝜔+𝜔𝑑𝜔𝜔2𝐶𝑚1,𝜇𝜃𝑎𝐶𝑚1,𝜇1𝑎+𝑎,(3.26) where 𝜃(𝑎2,1)(0,1),=(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=𝐶𝑎20𝑚1,𝜇1𝜔+𝜔𝑚1,𝜇𝑎𝜔+𝜔𝑎𝑑𝜔𝜔2𝐶𝑎20𝑚1,𝜇1𝜔+𝜔𝑚1,𝜇1𝑎𝑑𝜔𝜔2𝐶𝑚1,𝜇1𝑎10𝑚1,𝜇1𝜔+𝜔𝑑𝜔𝜔2𝐶𝑚1,𝜇1𝑎𝐶𝑚1,𝜇1𝑎+𝑎,(3.27) where =/21/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,𝑔𝑆𝜈𝜇,+(). 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 𝑆𝜇𝜈(×+).

Definition 3.4. Let there be given positive constants ,𝜈 and 𝜇. The space 𝑆𝜇𝜈,(×+) consists of all 𝜏𝐶(×+) such that ||||𝜏(𝑏,𝑎)𝐶𝑚1,𝜈𝑏𝑚1+𝑎1,𝜇1𝑎+𝑎,(𝑏,𝑎)×+,(3.28) for some 𝐶>0.
Then 𝑆𝜇𝜈(×+)=>0𝑆𝜇𝜈,(×+), and the topology is defined by the inductive limit: 𝑆𝜇𝜈×+=indlim>0𝑆𝜇𝜈,×+.(3.29)

Theorem 3.5. Let there be given 𝜈1 and 𝜇>0 and let 𝑓,𝑔𝑆𝜈𝜇,+(). Then, the wavelet transform 𝑊(𝑓,𝑔)𝑊𝑔𝑓 is a continuous map from 𝑆𝜈𝜇,+()×𝑆𝜈𝜇,+() into 𝑆𝑠𝜈(×+), where 𝑠=max{𝜈,𝜇}.

Proof. Since 𝑓 and 𝑔 are smooth, 𝑊𝑔𝑓(𝑏,𝑎)𝐶(×+). Proposition 2.8 implies that ||||𝑓(𝜔)𝐶𝑚1,𝑠1𝜔+𝜔,||||̂𝑔(𝜔)𝐶𝑚1,𝑠1𝜔+𝜔,𝜔>0,(3.30) for 𝑠=max{𝜈,𝜇} and for some positive constants and 𝐶.
By Propositions 3.2 and 3.3, it follows that there exist positive constants 𝐶 and such that ||𝑊𝑔||𝑓(𝑏,𝑎)𝐶𝑚1/2,𝜈𝑏𝑚1+𝑎1/2,𝑠1𝑎+𝑎,(𝑏,𝑎)×+,(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 𝑀𝑔 of function 𝜏𝑆𝜇𝜈(×+) with respect to the wavelet 𝑔𝐿1() given by 𝑀𝑔𝜏(𝑥)=0+𝑑𝑎𝑎𝜏1(𝑏,𝑎)𝑎𝑔𝑥𝑏𝑎𝑑𝑏,𝑥.(4.1)

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

Proposition 4.1. Let 𝜈1 and 𝜇>0. If 𝜏𝑆𝜇𝜈(×+) and if 𝑔 satisfies ||||𝑔(𝑥)𝐶𝑚1,𝜈(𝑥),𝑥,(4.2) for some positive constants and 𝐶, then ||𝑀𝑔||𝜏(𝑥)𝐶𝑚1,𝜈(𝑥),𝑥,(4.3) for some positive constants and 𝐶.

Proof. We have ||𝑀𝑔𝜏||(𝑥)0+𝑑𝑎𝑎||𝜏||1(𝑏,𝑎)𝑎|||𝑔𝑥𝑏𝑎|||𝑑𝑏𝐶0+𝑑𝑎𝑎2𝑚1,𝜈𝑏𝑚1+𝑎1,𝜇1𝑎+𝑎𝑚1,𝜈𝑥𝑏𝑎𝐼𝑑𝑏𝐶1(𝑥)+𝐼2(𝑥),𝑥,(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 (0,) we have 𝐼1(𝑥)10𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝑎2𝑚1,𝜈𝑏𝑚1+𝑎1,𝜈(𝑥)𝑚,𝜈(𝑏)𝑑𝑏,𝑥.(4.5) Since 𝑚1,𝜈𝑏𝑚1+𝑎,𝜈(𝑏)𝑑𝑏<,(4.6) when <21/𝜈(1+𝑎)1/𝜈 and since 10𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝑎2<,(4.7) it follows that 𝐼1(𝑥)𝐶𝑚1,𝜈(𝑥),𝑥,(4.8) for some 𝐶>0.
Next, we estimate 𝐼2. We have 𝐼2(𝑥)=1+𝑚1,𝜇1𝑎+𝑎𝑑𝑎1𝑎𝑚1,𝜈𝑏11+𝑎𝑎𝑚1,𝜈𝑥𝑏𝑎𝑑𝑏1+𝑚1,𝜇1𝑎+𝑎𝑑𝑎1𝑎𝑚1,𝜈𝑏12𝑎𝑎𝑚1,𝜈𝑥𝑏=2𝑎𝑑𝑏1+𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝐷𝑎𝑚1,𝜈𝑏2𝐷𝑎𝑚1,𝜈𝑥𝑏2=𝑑𝑏1+𝑚1,𝜇1𝑎+𝑎𝐷𝑎𝑚1,𝜈𝑏2𝐷𝑎𝑚1,𝜈𝑏2=(𝑥)𝑑𝑎1+𝑚1,𝜇1𝑎+𝑎𝐷𝑎𝑚1,𝜈𝑏2𝑚1,𝜈𝑏2(𝑥)𝑑𝑎,𝑥.(4.9)
By Lemma 3.1, 𝐼2(𝑥)𝐶1+𝑚1,𝜇1𝑎+𝑎𝐷𝑎𝑚1,𝜈(𝑥)𝑑𝑎,𝑥,(4.10) for every (0,). After the change of variables 𝑎=1/𝑎 and renaming 𝑎 to 𝑎, by the mean value theorem, we obtain 𝐼2(𝑥)10𝐶𝑚1,𝜇1𝑎+𝑎𝑚1,𝜈(𝑎𝑥)𝑑𝑎𝑎=𝐶𝑚1,𝜈(𝜃𝑥)10𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝑎,𝑥,(4.11) for some 𝜃(0,1). Therefore, there are 𝐶>0 and >0 such that 𝐼2(𝑥)𝐶𝑚1,𝜈(𝑥),𝑥, 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 𝑀𝑔𝜏 then the one given in [9].

Proposition 4.2. Let 𝜈 and 𝜇 be positive constants. If 𝜏𝑆𝜇𝜈(×+) and if 𝑔 satisfies ||||̂𝑔(𝜔)𝐶𝑚1,𝜇1𝜔+𝜔,𝜔>0,(4.12) for some positive constants and 𝐶, then |||𝑀𝑔|||𝜏(𝜔)𝐶𝑚1,𝜇1𝜔+𝜔,𝜔>0,(4.13) for some positive constants and 𝐶.

Proof. The Fourier transform of 𝑀𝑔𝜏 is given by 𝑀𝑔𝜏(𝜔)=0+𝑑𝑎𝑎𝜏(𝑏,𝑎)̂𝑔(𝑎𝜔)𝑒𝑖𝑏𝜔𝑑𝑏,𝜔>0.(4.14) Therefore, |||𝑀𝑔𝜏|||(𝜔)𝐶0+1𝑎𝑚1,𝜇1𝑎+𝑎𝑚1,𝜇1𝑎𝜔+𝑎𝜔𝑑𝑎𝑚1,𝜈𝑏1+𝑎𝑑𝑏𝐶1𝐼1(𝜔)+𝐼2(𝜔),𝜔>0,(4.15) where 𝐼1(𝜔)=101+𝑎𝑎𝑚1,𝜇1𝑎+𝑎𝑚1,𝜇1𝑎𝜔+𝐼𝑎𝜔𝑑𝑎,2(𝜔)=1+1+𝑎𝑎𝑚1,𝜇1𝑎+𝑎𝑚1,𝜇1𝑎𝜔+𝑎𝜔𝑑𝑎,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<𝜔1. This gives 𝐼1(𝜔)𝑚1,𝜇1𝜔102𝑎𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝑚1,𝜇121𝜔+𝜔],,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)𝐶𝑚1,𝜇1𝜃𝜔+𝜃𝜔102𝑎𝑚1,𝜇1𝑎+𝑎𝑑𝑎𝐶𝑚1,𝜇(𝜃𝜔)𝐶𝑚1,𝜇1𝜔+𝜔,𝜔(1,),(4.18) where 𝜃(0,1) and =(𝜃/2)1/𝜇.
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)2𝑚1,𝜇𝑎+1𝑎𝑚1,𝜇𝑎𝑎𝜔+𝜔𝑑𝑎102(𝑎)2𝑚1,𝜇𝑎+1𝑎𝑚1,𝜇𝑎1𝜔+𝜔𝑑𝑎=𝑚1𝜃1/𝜇,𝜇1𝜔+𝜔102(𝑎)2𝑚1,𝜇𝑎+1𝑎𝑑𝑎𝐶𝑚1,𝜇1𝜔+𝜔,𝜔(0,1),(4.19) where 𝜃(0,1) and =𝜃1/𝜇.
In a similar way, as in the proof of Proposition 3.3, it follows that 𝐼2(𝜔)𝐶𝑚121/𝜇,𝜇1𝜔+𝜔,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,𝜈 is not submultiplicative on the “𝑎𝑥+𝑏”-group.

Theorem 4.3. Let 𝜈1 and 𝜇>0. The inverse wavelet transform 𝑀(𝑔,𝜏)𝑀𝑔𝜏 is a continuous map from 𝑆𝜈𝜇,+()×𝑆𝜇𝜈(×+) into 𝑆𝜈𝑠,+(), where 𝑠=max{𝜈,𝜇}.
Moreover, if 𝑔0, then for any 𝑓𝑆𝜈𝑠,+() the following inversion formula holds: 1𝑓(𝑥)=𝑐𝑔0𝑑𝑎𝑎𝑊𝑔1𝑓(𝑏,𝑎)𝑎𝑔𝑥𝑏𝑎𝑑𝑏,(4.21) where 𝑐𝑔=0|̂g(𝜔)|2(𝑑𝜔/𝜔).

Proof. For the first part of Theorem 4.3, it is enough to observe that Proposition 2.8 implies that ||||̂𝑔(𝜔)𝐶𝑚1,𝑠1𝜔+𝜔,𝜔>0,(4.22) for some positive constants 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 |||𝑀𝑔|||𝜏(𝜔)𝐶𝑚1,𝑠1𝜔+𝜔𝐶𝑚1,𝑠(𝜔),𝜔>0.(4.23)
From the the proof of Proposition 2.8, it follows that 𝑔 satisfies the admissibility condition: ||||0<0||||̂𝑔(𝜔)2𝑑𝜔𝜔||||<.(4.24) Since 1𝑐𝑔0𝑑𝑎𝑎𝑊𝑔1𝑓(𝑏,𝑎)𝑎𝑔𝑥𝑏𝑎1𝑑𝑏=𝑐𝑔𝑀𝑔𝑊𝑔𝑓(𝑥),𝑥,(4.25) and by Theorem 3.5, we have that 𝑊𝑔𝑓𝑆𝑠𝜈(×+), from the first part of the proof it follows that 𝑀𝑔(𝑊𝑔𝑓)𝑆𝜈𝑠,+().
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 𝑆𝜈𝑠,+() 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 >0 and 𝐶>0 such that (i)|𝑊𝑔𝑓(𝑏,𝑎)|𝐶𝑒|𝑏|1/𝜈,(𝑏,𝑎)×(0,𝑀]; (ii)|𝑊𝑔𝑓(𝑏,𝑎)|𝐶𝐷𝑎(𝑒||1/𝜈)(𝑏),(𝑏,𝑎)×[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 ,𝜈, and 𝜇. The space 𝑍𝜇𝜈,(×+) consists of all 𝜏𝐶(×+) such that ||||𝜏(𝑏,𝑎)𝐶𝑚1,𝜈(𝑏)𝑚1,𝜇1𝑎+𝑎,(𝑏,𝑎)×+,(5.1) for some 𝐶>0.
Then 𝑍𝜇𝜈(×+)=>0𝑍𝜇𝜈,(×+), and the topology is given by the inductive limit: 𝑍𝜇𝜈×+=indlim>0𝑍𝜇𝜈,×+.(5.2)
Next, we reformulate results from Sections 3 and 4 in terms of 𝑍𝜇𝜈(×+) since we believe that this might be of independent interest.

Theorem 5.2. Let 𝜈>1 and 𝜇>0, let 𝑓,𝑔𝑆𝜇,+𝜈𝜀(), with 𝜀>0 chosen such that 𝜈𝜀1. Then, the wavelet transform 𝑊(𝑓,𝑔)𝑊𝑔𝑓 is a continuous map from 𝑆𝜇,+𝜈𝜀()×𝑆𝜇,+𝜈𝜀() into 𝑍𝑠𝜈(×+), 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 𝑒𝑁||1/(𝜈𝜀)𝐶𝑒𝑀||1/𝜈,(5.3) wherefrom ||𝑊𝑔||𝐶𝑓(𝑏,𝑎)𝑒|𝑏/(1+𝑎)|1/(𝜈𝜀)𝐶𝑒𝑘|𝑏|1/𝜈,(𝑏,𝑎)×+,(5.4) for arbitrary 𝑘>0. The theorem is proved.

The inverse wavelet transform 𝑀𝑔 of 𝜏𝑍𝜇𝜈(×+) with respect to the wavelet 𝑔𝐿1() given by 𝑀𝑔𝜏(𝑥)=0+𝑑𝑎𝑎𝜏1(𝑏,𝑎)𝑎𝑔𝑥𝑏𝑎𝑑𝑏,𝑥.(5.5)

Propositions 4.1 and 4.2 still hold true with 𝑆𝜇𝜈(×+) replaced by 𝑍𝜇𝜈(×+), while Theorem 4.3 takes the following form.

Theorem 5.3. Let 𝜈1 and 𝜇>0. The inverse wavelet transform 𝑀(𝑔,𝜏)𝑀𝑔𝜏 is a continuous map from 𝑆𝜈𝜇,+()×𝑍𝜇𝜈(×+) into 𝑆𝜈𝑠,+(), where 𝑠=max{𝜈,𝜇}.
Moreover, if 𝑓,𝑔𝑆𝜇,+𝜈𝜀() for some 𝜀>0 such that 𝜈𝜀>1 and 𝑔0, then 1𝑓(𝑥)=𝑐𝑔0𝑑𝑎𝑎𝑊𝑔1𝑓(𝑏,𝑎)𝑎𝑔𝑥𝑏𝑎𝑑𝑏,(5.6) where 𝑐𝑔=0|̂𝑔(𝜔)|2(𝑑𝜔/𝜔).


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