Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 469341, 12 pages

doi:10.1155/2012/469341

Research Article

Reproducing Kernels and Variable Bandwidth

R. Aceska¹ and H. G. Feichtinger²

¹Faculty of Mechanical Engineering, Ss. Cyril and Methodius University, Karposh II b.b., 1000 Skopje, Macedonia

²Faculty of Mathematics, NuHAG, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Received 20 June 2012; Accepted 27 August 2012

Academic Editor: Baoxiang Wang

Copyright © 2012 R. Aceska and H. G. Feichtinger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We show that a modulation space of type () is a reproducing kernel Hilbert space (RKHS). In particular, we explore the special cases of variable bandwidth spaces Aceska and Feichtinger (2011) with a suitably chosen weight to provide strong enough decay in the frequency direction. The reproducing kernel property is valid even if () does not coincide with any of the classical Sobolev spaces because unbounded bandwidth (globally) is allowed. The reproducing kernel will be described explicitly.

1. Motivation and Preliminaries

Classical Sobolev spaces [1, 2] are defined by weighting on the frequency side with

It is submultiplicative for and defines a reproducing kernel Hilbert space (RKHS), whenever [3]. Knowing that modulation spaces , defined with respect to (1.1), coincide with (Prop.  11.3.1, [4]) and are therefore RKHSs, it is reasonable to ask if a reproducing kernel (RK) for a modulation space with respect to a more general weight exists. Even for the classical case, we have the Sobolev embedding condition to be satisfied, so cannot be arbitrary.

Further on, we will explore what happens to the RK when the weight mildly varies from (1.1). Or, more precisely, we check if spaces of variable bandwidth [5] (VB-spaces for short) are in fact RKHSs and what is the specific design of the respective RK.

By the Riesz representations theorem [6], in a Hilbert space with inner product , we seek for the existence of a function , , that satisfies the reproducing property

for each . Then it also holds , , .

The work is here organized as follows: after reviewing basic time-frequency tools and modulation spaces, in the next section, we adapt the VB concept to the Sobolev-type weights. Further on, we calculate the RK for a general modulation space. Then we observe the special examples with respect to the standard weight (1.1) and the respective inner product and a VB weight with respect to a continuous and a constant bandwidth.

Because it simplifies our proofs, we use a normalized Schwartz window , that is, . We denote by the space of tempered distributions, while and denote the space of test functions with compact support and its dual space.

1.1. Fourier Transform, Sobolev Space

Classical Fourier analysis employs two complementary representations to describe functions: the function itself (temporal behavior) and its Fourier transform

defined on , which describes the frequency behavior. Plancherel's theorem gives the possibility of extending to a unitary operator on and satisfies Parseval's formula . Its inverse transform equals its adjoint operator and , where is the reflection . It holds

where is the translation operator.

As a rule of thumb, smoothness of implies decay of and vice versa. Thus it makes sense to work with the following class of Sobolev spaces, the so-called Bessel potential spaces:

For , this is a subspace of consisting of smooth functions. For (negative smoothness), one has and such spaces may include discrete measures and distributions. The respective inner product is of form

where is the -inner product and is as in (1.1).

In terms of a generalized inner product, the inverse Fourier transform can be written as . If we want the representation of via (1.6), we have

Thus , so . After applying (1.4), we derive the reproducing kernel for the Sobolev space with respect to the classic inner product (1.6) to be

provided [3].

The classical Sobolev spaces as defined above in (1.5) have an alternative description in the context of modulation spaces, where one has to use the same weight as a function of two variables (Example 1.1). Thus, it makes sense to calculate the RK in the general case.

1.2. Weights

A weight is a positive function on . By , we denote a positive, even, submultiplicative weight such that , which satisfies the inequality , for all , . A positive, even weight function on is called -moderate if it satisfies for some

Two weights and are called equivalent if one has for some

Commonly used weights are the (equivalent) submultiplicative weights of polynomial type and , because one has

It is easy to show that, if the moderate weight (with respect to ) is equivalent to a weight , then is also moderate with respect to the same weight [5]. For more information on the use of weights in time-frequency analysis, we refer the reader to [4, 7].

1.3. Short-Time Fourier Transform

By a time-frequency shift of a function , we mean , for any pair . For convenience, we will sometimes use the following notation for a function (depending on and ) on the TF-plane: , .

The short-time Fourier transform (STFT for short notation) of a function with respect to a window -function is

for all and it holds . Compared to (1.3), the STFT is a significant improvement as it is a joint time-frequency representation; by its structure it is in fact a localized Fourier transform.

The definition of can be generalized to larger classes, whenever the inner product in (1.12) is well defined (for instance: and ; as this choice gives us greatest freedom in choosing the function at hand, we will only use Schwartz atoms in our work). In fact, it is enough that and belong to time-frequency shift-invariant, mutually dual spaces. As an example, if , then it is possible to calculate the STFT of the delta distribution by

The inversion formula for the STFT

is well defined in the weak sense for all and windows such that . Its calculation via Riemannian sums is explored in [8]. Written as vector-valued integrals, we have the adjointness relation for and . is a well-defined operator from a weighted (mixed-norm) space over the TF-plane to the corresponding modulation space.

1.4. Modulation Spaces

Modulation spaces theory is a special example of the much wider coorbit theory [9–11], which covers the case of solid, translation-invariant spaces. We comprise here known properties and facts about modulation space, following mostly [4]. Given a fixed non-zero window , a -moderate weight on and , , the modulation space consists of all tempered distributions for which has finite weighted mixed -norm

(with the usual adjustment for , ). If , we write instead of and if then we write instead of . A special example is Feichtinger's algebra . It is sometimes useful to calculate with and , in particular when we want to go beyond -functions.

Example 1.1. Interesting examples of modulation spaces ([4], page 232) are

(a) with (b)in particular (c) when (d), which coincides with , .

Given any moderate weight , the space is a time-frequency shift-invariant Hilbert space, with inner product

The norm deriving from this inner product is independent of the choice of the used window that is, two nonzero windows and produce equivalent norms, and there exists a constant such that for all it holds . Similarly, equivalent weights and define the same modulation space.

2. Reproducing Kernel with respect to a -Moderate Weight

Given a -moderate weight , we seek a two-dimensional function such that for all , , it holds , that is

We compare to (1.14) and conclude that , that is,

Theorem 2.1. The reproducing kernel for defined with respect to a -moderate weight is

For -fixed,

Proof. Let be fixed, we have

Corollary 2.2. Let and let be equivalent to with equivalence constant . Denote the respective reproducing kernels with and .

If then

Proof . Instead of using to reconstruct the function at point in the modulation space defined with respect to , we use to obtain the approximation as follows:

Then it holds , which is equivalent to (2.5).

For instance, if the equivalence constant between weights and is , then for all , it holds that .

2.1. Reproducing Kernel for the Sobolev Space

In analogy to (1.8), we calculate the RK for the modulation space , equivalent to , defined with respect to weight (1.1). Since we now employ the inner product (2.1), the RK of depends on the used analysis window .

Let the RK function be denoted by , then (2.2) implies that , and by the pointwise version of (1.14)

We use (1.4) to obtain the RK formula (2.8).

Corollary 2.3. Let and . The reproducing kernel of is

where and

Proof. From (2.3), it follows

The last equality indicates that the condition is necessary.

Note 1. is decaying fast for a fast-decaying window . For instance, if we have

which decays fast the further away are and . In the discrete setting, is a diagonal-like matrix with fast off-diagonal decay.

Note 2. Compared to (1.8), the reproducing kernel in (2.8) has an extra multiplier , which varies for different windows.

3. Customized Weights on the Time-Frequency Plane

The idea of variable bandwidth was first described in Slepian's talk [12]; implicitly, it was studied in [13, 14] and many others, while explicitly it was explored in [15–18]. Here we work with the VB concept we have developed in [5].

Let the function describe the time-varying broadness of a strip in the time-frequency plane , that is,

We call the set a strip with variable bandwidth (VB strip).

The vertical distance function at point is given by

In [5], we defined a variable bandwidth weight of order on the time-frequency plane (VB weight) by

and proved it is a moderate weight. In this paper, we work with a Sobolev-type weight

which is equivalent to (3.3). The most simple example is a weight with respect to a constant bandwidth for some fixed .

Notice that, whenever we choose in the last equation, we get the standard Sobolev weight .

We choose to work with in (3.4) for the following reasoning: when applying weight (3.4) on the time-frequency plane, we add graded weight to the exterior of . It can be shown that, for a well-localizing window , if the weighted is integrable for some , then is decaying faster then a polynomial of order outside the strip. In other words, this weight is giving us the opportunity to locally describe the STFT decay.

It is a simple exercise to prove that (3.4) is moderate; all one needs is the equivalence inequality with respect to (3.3), which is analogue to the equivalence of the two submultiplicative weights in (1.11). Thus, with a simple adaptation of Proposition 3.1 from [5], the following result holds.

Proposition 3.1. Let , take to be a nonnegative function on and define weight (3.4) for given as in (3.2). If for some the boundary function satisfies

then the weight is moderate with respect to

That is, it holds for all , , , .

As a consequence, is moderate if is bounded (since is then satisfying (3.6), known as the Lipschitz condition). This corresponds to in Proposition 3.1, thus the bandwidth function is bounded and consequently, the controlling weight depends only on the frequency variable .

Corollary 3.2. If is a bounded function, then the weight , given by (3.4), is moderate with respect to .

Due to the uncertainty principle it does not make sense to talk of the bandwidth at a given point, nor to try to describe rapid changes of local bandwidth. Accordingly, as seen in [5], the concept of variable bandwidth must be designed with some built-in robustness, and small local changes of the parameters should not effect the resulting spaces.

Proposition 3.3. Let for all , and let generate a moderate weight (3.4). Then generates a moderate weight , equivalent to , provided .

Proof. We work here with : let for all . The equivalence inequality is trivially satisfied on as both weights have value . Let . Then , and we have

If , then both weights have nontrivial values. Using this estimate

which holds true whenever , we derive that

Then it holds .

Equivalent weights are moderate simultaneously, thus is moderate.

If we apply Proposition 3.3 a finite number for shifting the constant bandwidth in (3.5) toward , we have the following Corollary.

Corollary 3.4. A VB weight , defined with respect to a bounded bandwidth , is equivalent to .

3.1. Banach Spaces of Variable Bandwidth Functions

Using the moderate weights defined with respect to a variable bandwidth strip (3.1), we can proceed to the definition of functions with variable bandwidth, using the tools from Subsection 1.4. We have seen that VB weights provide for a certain flexibility, that is, the precise knowledge of the bandwidth is not necessary as finite changes give equivalent weights. This will provide for equivalent norms on the function spaces level.

Definition 3.5. Let be defined on and fix . We say that is a function with variable bandwidth if it belongs to a modulation space for some and a VB weight , related to and defined as in (3.4).

We will call the space a variable bandwidth space and refer to it by VB space when parameters are known and is fixed up to equivalence. As usual, if we write instead of . The modulation space (and VB space) norm in this paper is marked with or , if necessary. Since weights (3.3) and (3.4) are equivalent, no matter which of the two weights we use, they define one VB space. Thus the properties of the spaces defined in Definition 3.5 are the same as described in [5].

4. Special Cases of VB Reproducing Kernels

Corollary 4.1. The reproducing kernel for the VB-space , defined with respect to (3.4), is and the norm of depends on the bandwidth function

Proof. Due to (2.3) and the symmetry properties of (3.4), it holds

Our a priori assumption is that we work with a normalized window that is, , so (4.1) holds.

A similar split of integration, as seen in the proof of Corollary 2.3, is possible for a constant bandwidth weight (3.5). Locally, depends only on .

Corollary 4.2. The RK for , where is as in (3.5), is

where is defined in (2.9).

Proof. By (2.2), we have . Recall (3.5) and its symmetry property; therefore, we can use the notation to describe the weight , for and otherwise and obtain

Acknowledgment

The authors would like to thank Peter Balasz for the fruitful discussions in the early writing stages of this paper.

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