Progressive Gelfand-Shilov Spaces and Wavelet Transforms
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.
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 , 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 which consists of functions in whose Fourier transforms are supported by the right half-plane. It is well known that Hardy spaces ,, play fundamental role in signal processing, cf. [5–7]. Analytic signal , associated to a real-valued signal by , where is the Hilbert transform of , belongs to , 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 . 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 ,, with subexponential weights of the form ,, . 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 ,, and a weight of the form , , . The Haar module type weights are submultiplicative with respect to the action of the “”-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 . Here, we give explicit calculations for subexponential weights and give explanations concerning more general situations whenever convenient.
Gelfand-Shilov spaces were introduced in  (and called there the spaces of type ) 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  it is shown that these spaces can be used to construct a noncommutative quantum field theory.
Our results can be compared to  where one can find certain global type results. The analyzing wavelets in  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, , are Lebesgue spaces, denotes the dual pairing of and , and the Fourier transform of is normalized to be By we denote the dilation . We adopt the following notation used in the study of pseudo-differential operators (cf. [17, 20]): .
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 are introduced in Definition 2.1. The range of the wavelet transform over such spaces is contained in , see Definition 3.4, and in , see Definition 5.1, the latter being introduced for the analysis of decay of wavelet transform with respect to variables and separately, for example, Section 5.
2. Progressive Gelfand-Shilov Spaces
Let , and be positive constants such that . By we denote the Banach space which consists of all such that the norm is finite.
The Gelfand-Shilov space is given by and its topology is given by the inductive limit:
Definition 2.1. Let , and , be positive constants such that . The space consists of all such that and such that (2.1) is finite.
The progressive Gelfand-Shilov space is given by and its topology is given by the inductive limit:
Obviously, is a closed subspace of .
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 . For the projective limits of , , we refer to [21, 22].
Note also that the decay of can be controlled by some other choice of sequences ,, instead of the Gevrey sequences and used in (2.1). We refer to [1, 21–23] for the usual conditions on and .
For the sake of the clarity of the exposition, in this paper, we observe the inductive limits and the Gevrey sequences and , which is not an essential restriction.
It can be shown that multiplication by and differentiation are continuous operations in and that the same is true for translation and dilation by a positive factor, for example [2, Chapter IV]. On the other hand, is not closed under modulations. This implies that Short-time Fourier transform, an important tool in local time-frequency analysis (see  for the definition and basic properties), is not appropriate for the analysis of . 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 and . Then, the following conditions are equivalent. (a)There exists such that for some . (b)There exists such that
Lemma 2.4. Let and . There exist such that
Remark 2.5. When and by Proposition 2.3, it directly follows that the following conditions are equivalent. (a)There exists such that for some .(b)There exists such that
Next, we recall the characterization of Gelfand-Shilov spaces.
Theorem 2.6. Let and be positive constants such that and let . Then, the following conditions are equivalent,(a). (b)There exist and (which depend only on ) such that (c)There exist and (which depend only on ) such that (d)There exist and (which depend only on ) such that (e)There exist and (which depend only on ) such that
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 are used. Another proof which follows from direct computations is given in [17, Theorem 6.1.6.], see also [21, 22].
Moreover, elements in satisfy an additional localization property, which is sometimes called "strip localization." We refer to  for the definition of polynomial strip localized progressive functions. Let us show that elements from are almost exponentially strip localized.
Proposition 2.8. Let and be positive constants such that and let satisfy , that is, . Then, , where .
Proof. Note that implies ,, that is, and when , for every . By Taylor's formula, it follows that where we have used Theorem 2.6(d) and , for example [26, page 59]. Therefore, which, together with Theorem 2.6(c), implies that there exist and such that By Remark 2.5 there exist and such that as claimed.
3. Wavelet Transform
In this section, we define the wavelet transform and study its decay and continuity properties.
The wavelet transform of with respect to the wavelet is given by and can be extended to any dual pair of spaces provided that duality is meaningful.
In particular, is well defined when . Theorem 2.6 (e) suggests to observe the behavior of and in the configuration and in the frequency domain separately in the study of when .
We introduce following notation for the subexponential weight function: where .
When and the weight is radial, submultiplicative and subconvolutive. Recall, a nonnegative function is called subconvolutive if and for some constant .
When , the following lemma follows from [12, Lemma 7.1], see also , 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 here.
Lemma 3.1. Let there be given , and functions and such that for some positive constant . Then, for some and for every such that .
Proof. By the elementary inequality ,,, it follows that for a given we have Therefore, By the assumption (3.4), it follows that where we choose such that . The proof is complete.
Proposition 3.2. Let there be given , and let and satisfy (3.4) for some . Then, there exist such that
Proof. When the so-called “voice transform” is just the convolution between and , so by Lemma 3.1 we have for every such that . In particular, when , we obtain for some . If , then When , by the symmetry property of the wavelet transform: (cf. [9, (2.0.1)]) and by the first part of the proof we have: The proof is complete.
Proposition 3.2 can be compared to , where an extra factor appears, and where the different technique is used for the proof. Note that (sub)exponential growth of leads to the convergent integral, whereas the same type of estimate for a weight of polynomial growth of the form gives a divergent integral:
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 and let and satisfy for some positive constant . Then, there exist and such that
Proof. By the Parseval's formula, it follows that
The change of variables gives
When , we have where . Therefore, it is enough to estimate when .
We split in three parts:
When and , we have , wherefrom Since it follows that
Next, we make the change of variables: in and and then rename into .
By the mean value theorem, where and we have used inequalities and (3.24).
It remains to estimate . We have where and we have used inequalities and (3.24).
This completes the proof.
We end the section with a continuity result when . We refer to [9, Theorem 19.0.1] for the case of progressive functions of polynomial decay, and to  for the (Gelfand-Shilov) spaces of type . First, we introduce the space .
Definition 3.4. Let there be given positive constants and . The space consists of all such that
for some .
Then , and the topology is defined by the inductive limit:
Theorem 3.5. Let there be given and and let . Then, the wavelet transform is a continuous map from into , where .
Proof. Since and are smooth, . Proposition 2.8 implies that
for and for some positive constants and .
By Propositions 3.2 and 3.3, it follows that there exist positive constants and such that which proves the continuity.
Note that and in Theorem 3.5 belong to the same progressive Gelfand-Shilov space, while in  different decay conditions are imposed on and . Since the analyzing wavelet in  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 of function with respect to the wavelet given by
Proposition 4.1. Let and . If and if satisfies for some positive constants and , then for some positive constants and .
Proof. We have
where in the integration with respect to is taken over and in over .
By (3.6), it follows that for any we have Since when and since it follows that for some .
Next, we estimate . We have
By Lemma 3.1, for every . After the change of variables and renaming to , by the mean value theorem, we obtain for some . Therefore, there are and such that , and (4.3) is proved.
Proposition 4.2. Let and be positive constants. If and if satisfies for some positive constants and , then for some positive constants and .
Proof. The Fourier transform of is given by
and we have made the change of variables .
We use similar arguments as in proof of Proposition 3.3 to estimate when . This gives where we have used the inequality ,.
When by the mean value theorem, we obtain where and .
It remains to estimate .
When , by change of variables and by the mean value theorem, it follows that where and .
In a similar way, as in the proof of Proposition 3.3, it follows that 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 is not submultiplicative on the “”-group.
Theorem 4.3. Let and . The inverse wavelet transform is a continuous map from into , where .
Moreover, if , then for any the following inversion formula holds: where .
Proof. For the first part of Theorem 4.3, it is enough to observe that Proposition 2.8 implies that
for some positive constants and , and for .
The continuity now follows from Propositions 4.1 and 4.2 and Theorem 2.6(e), since
From the the proof of Proposition 2.8, it follows that satisfies the admissibility condition: Since and by Theorem 3.5, we have that , from the first part of the proof it follows that .
The proof is now the same as the proof of [9, Theorem 14.0.2] if we take to be its own reconstruction wavelet and observe 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 there exist and such that (i); (ii).
Note that those estimates imply that we are not able to separate variables and 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 and separately. Therefore, we have the following definition.
Definition 5.1. Let there be given positive constants , and . The space consists of all such that
for some .
Then , and the topology is given by the inductive limit:
Next, we reformulate results from Sections 3 and 4 in terms of since we believe that this might be of independent interest.
Theorem 5.2. Let and , let , with chosen such that . Then, the wavelet transform is a continuous map from into , where .
Proof. It follows from the proof of Theorem 3.5 and the fact that for every positive , and there is a constant such that wherefrom for arbitrary . The theorem is proved.
The inverse wavelet transform of with respect to the wavelet given by
Theorem 5.3. Let and . The inverse wavelet transform is a continuous map from into , where .
Moreover, if for some such that and , then where .
The research was supported by MPN of Serbia, Project nos. 174024, III 44006 and by PSNTR Project no. 114-451-2167.
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