Reproducing Kernels and Variable Bandwidth

We show that a modulation space of type M2 m R 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 ifM2 m R does not coincide with any of the classical Sobolev spaces because unbounded bandwidth globally is allowed. The reproducing kernel will be described explicitly.


Motivation and Preliminaries
Classical Sobolev spaces H 2 s R 1, 2 are defined by weighting on the frequency side with It is submultiplicative for s ≥ 0 and defines a reproducing kernel Hilbert space RKHS , whenever s > 1/2 3 .Knowing that modulation spaces M 2 v o s R , defined with respect to 1.1 , coincide with H 2 s R Prop.11.3.1, 4 and are therefore RKHSs, it is reasonable to ask if a reproducing kernel RK for a modulation space M M 2 m R with respect to a more general weight m exists.Even for the classical case, we have the Sobolev embedding condition s > 1/2 to be satisfied, so m 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 M with inner product •, • M , we seek for the existence of a function Φ y ∈ M, y ∈ R, that satisfies the reproducing property for each f ∈ M. Then it also holds Φ t y Φ t , Φ y M , t, y ∈ R. 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 g ∈ S R , that is, g 2 1.We denote by S the space of tempered distributions, while D and D denote the space of test functions with compact support and its dual space.

Fourier Transform, Sobolev Space
Classical Fourier analysis employs two complementary representations to describe functions: the function f itself temporal behavior and its Fourier transform defined on L 1 R , which describes the frequency behavior.Plancherel's theorem gives the possibility of extending F to a unitary operator on L 2 R and satisfies Parseval's formula f, g f, g .Its inverse transform equals its adjoint operator and F −1 IF, where I is the reflection If t f −t .It holds where T y is the translation operator.
As a rule of thumb, smoothness of f implies decay of f and vice versa.Thus it makes sense to work with the following class of Sobolev spaces, the so-called Bessel potential spaces: For s ≥ 0, this is a subspace of L 2 R consisting of smooth functions.For s < 0 negative smoothness , one has L 2 ⊆ H 2 s and such spaces may include discrete measures and distributions.The respective inner product is of form where •, • is the L 2 -inner product and v v o s is as in 1.1 .
In terms of a generalized inner product, the inverse Fourier transform can be written as f y f, e −2πiy• .If we want the representation of f via 1.6 , we have Thus F Φ y v 2 e −2πiy• , so Φ y F −1 v −2 e −2πiy• .After applying 1.4 , we derive the reproducing kernel for the Sobolev space H 2 s R with respect to the classic inner product 1.6 to be 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.

Weights
A weight is a positive function on R 2 .By v, we denote a positive, even, submultiplicative weight such that v 0 1, which satisfies the inequality v z 1.9 Two weights m 1 and m 2 are called equivalent if one has for some C > 0 Commonly used weights are the equivalent submultiplicative weights of polynomial type 1 |ω| s and 1 ω 2 s/2 , s ≥ 0 because one has It is easy to show that, if the moderate weight m o with respect to v is equivalent to a weight m, then m is also moderate with respect to the same weight v 5 .For more information on the use of weights in time-frequency analysis, we refer the reader to 4, 7 .

Short-Time Fourier Transform
By a time-frequency shift of a function f, we mean M ω T x f t : e 2πit•ω f t − x , for any pair x, ω ∈ R 2 .For convenience, we will sometimes use the following notation for a function depending on g and y ∈ R on the TF-plane: π * g y x, ω : M ω T x g y , x, ω ∈ R.
The short-time Fourier transform STFT for short notation of a function f ∈ L2 R with respect to a window L 2 -function g is for all x, ω ∈ R 2 and it holds V g f 2 f 2 g 2 .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 V g f can be generalized to larger classes, whenever the inner product in 1.12 is well defined for instance: f ∈ S R and g ∈ S R ; 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 g and f belong to time-frequency shift-invariant, mutually dual spaces.As an example, if g ∈ D, then it is possible to calculate the STFT of the delta distribution δ y ∈ D by The inversion formula for the STFT is well defined in the weak sense for all f ∈ L 2 R and windows γ, g such that γ, g / 0. Its calculation via Riemannian sums is explored in 8 .Written as vector-valued integrals, we have the adjointness relation is a well-defined operator from a weighted mixed-norm space over the TF-plane to the corresponding modulation space.

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 g ∈ S, a v-moderate weight m on R 2 and 1 ≤ p, q < ∞, the modulation space M p,q m R consists of all tempered distributions f ∈ S R for which V g f has finite weighted mixed L p,q -norm with the usual adjustment for p, q ∞ .If p q, we write M Given any moderate weight m, the space M 2 m R 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 g and g 0 produce equivalent norms, and there exists a constant c > 0 such that for all . Similarly, equivalent weights m 1 and m 2 define the same modulation space.

Reproducing Kernel with respect to a v-Moderate Weight m
Given a v-moderate weight m, we seek a two-dimensional function

2.1
We compare to 1.14 and conclude that π * g y m 2 • V g Φ y , that is, Theorem 2.1.The reproducing kernel for M 2 m R defined with respect to a v-moderate weight m is For y-fixed,

2.3
Proof.Let y ∈ R be fixed, we have

2.4
Corollary 2.2.Let f ∈ M 2 m 1 R and let m 2 be equivalent to m 1 with equivalence constant c.Denote the respective reproducing kernels with Φ 1 and Φ 2 . If Proof .Instead of using Φ 1,y to reconstruct the function f at point y in the modulation defined with respect to m 1 , we use m 2 to obtain the approximation f as follows:

2.6
Then it holds c −2 V g f, π * g y ≤ f y ≤ c 2 V g f, π * g y , which is equivalent to 2.5 .
For instance, if the equivalence constant between weights m 1 and m 2 is c 1.05, then for all f ∈ M 2 m 1 , it holds that 0.907f y ≤ f y ≤ 1.1025f y .

Reproducing Kernel for the Sobolev Space
In analogy to 1.8 , we calculate the RK for the modulation space M 2 v R , equivalent to H 2 v R , defined with respect to weight 1.1 .Since we now employ the inner product 2.1 , the RK of M 2 v R depends on the used analysis window g.Let the RK function be denoted by Φ y t Φ t, y , then 2.2 implies that Φ y V * g v −2 π * g y , and by the pointwise version of 1.14 Φ t, y 1 1 ω 2 s e −2πiωy g y − x e 2πiωt g t − x dx dω e 2πiω t−y 1 ω 2 s dω g y − x g t − x dx.

2.7
We use 1.4 to obtain the RK formula 2.8 .

2.10
The last equality indicates that the condition s > 1/2 is necessary.

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 b x ≥ 0 describe the time-varying broadness of a strip ST b in the time- We call the set ST b a strip with variable bandwidth VB strip .
The vertical distance function d b at point z x, ω ∈ R 2 is given by In 5 , we defined a variable bandwidth weight of order s 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 a > 0 for some fixed s > 0.
Notice that, whenever we choose a ≡ 0 in the last equation, we get the standard Sobolev weight 1 |ω| 2 s/2 .We choose to work with s > 0 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 ST b .It can be shown that, for a well-localizing window g, if the weighted V g f is integrable for some s > 0, then V g f is decaying faster then a polynomial of order s 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 Proposiiton 1 from 5 , the following result holds.
As a consequence, m b is moderate if b is bounded since b is then satisfying 3.6 , known as the Lipschitz condition .This corresponds to k 0 in Proposition 3.1, thus the bandwidth function b is bounded and consequently, the controlling weight depends only on the frequency variable ω.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.

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.

m
and if m 1 then we write M p,q instead of M p,q Example 1.1.Interesting examples of modulation spaces 4 , page 232 are a

Proposition 3 . 1 .
Let s > 0, take b to be a nonnegative function on R and define weight 3.4 for d b given as in 3.2 .If for some k ≥ 0 the boundary function b satisfies ∀x, y ∈ R b x − b y ≤ k x − y , 3.6 then the weight m b is moderate with respect to

Corollary 3 . 2 .
If b is a bounded function, then the weight m b , given by 3.4 , is moderate with respect to v o s x, ω 1 |ω| 2 s/2 .

Proposition 3 . 3 .
Let |h x | < c for all x ∈ R, and let b ≥ 0 generate a moderate weight m b 3.4 .Then b h generates a moderate weight m b h , equivalent to m b , provided b h ≥ 0. Proof.We work here with c √ 2/2: let 0 ≤ h x < √ 2/2 for all x ∈ R. The equivalence inequality is trivially satisfied on ST b h as both weights have value 1.Let b x < |ω| ≤ b x h x .Then |ω| − b x ≤ h x , m b h x, ω 1 and we have

9 which holds true whenever h x < √ 2 / 2 ,
then both weights have nontrivial values.Using this estimate 0 ≤ |ω| − b x − 2h x 2 1 − 2h 2 x , 3.we derive that 1 |ω| − b x 2 ≤ 2 1 |ω| − b x − h x 2 .3.10 Then it holds m b h x, ω ≤ m b x, ω ≤ 2 s/2 m b h x, ω .Equivalent weights are moderate simultaneously, thus m b h is moderate.If we apply Proposition 3.3 a finite number for shifting the constant bandwidth in 3.5 toward 0, we have the following Corollary.Corollary 3.4.A VB weight m b , defined with respect to a bounded bandwidth b, is equivalent to 1 |ω| 2 s/2 .

Proof. By 2 . 2 ,e 0 e
we have Φ y t g t − x g y − x e 2πiω t−y /m 2 a x, ω dω dx.Recall 3.5 and its symmetry property; therefore, we can use the notation v o s to describe the weightv o s x, ω 1 ω 2 s/2 , for x > 0 and v o s 2πiω t−y 1 −ω − a 2 −s dω e 2πia t−y − e −2πia t−y 2πi t − y e 2πia t−y ∞ 2πiξ t−y 1 ξ 2 −s dξ e −2πia t−y ∞ 0 e −2πiξ t−y 1 ξ 2 −s dξ e 2πia t−y − e −2πia t−y 2πi t − y e −2πia t−y T y F v o s −2 t e 2πia t−y T y F −1 v o s −2 t .4.4