# Reproducing Kernels and Variable Bandwidth

Authors: R. Aceska; H. G. Feichtinger
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We show that a modulation space of typeMm2 (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 if Mm2
(ℝ) does not coincide with any of the classical Sobolev spaces because unbounded bandwidth (globally) is allowed. The reproducing kernel will be described explicitly.


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## Body

## 1. Motivation and Preliminaries

Classical Sobolev spacesℋs2(ℝ) [1, 2] are defined by weighting on the frequency side with
(1.1)vso(ω)=(1+ω2)s/2.
It is submultiplicative for s≥0 and defines a reproducing kernel Hilbert space (RKHS), whenever s>1/2 [3]. Knowing that modulation spaces Mvso2(ℝ), defined with respect to (1.1), coincide with ℋs2(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=Mm2(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 Φy∈ℳ, y∈ℝ, that satisfies the reproducing property(1.2)f(y)=〈f,Φy〉ℳ
for each f∈ℳ. Then it also holds Φt(y)=〈Φt,Φy〉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 windowg∈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.


### 1.1. Fourier Transform, Sobolev Space

Classical Fourier analysis employs two complementary representations to describe functions: the functionf itself (temporal behavior) and its Fourier transform
(1.3)ℱ(f)(ω)=f^(ω)=∫ℝ‍f(t)e-2πiωtdt
defined on L1(ℝ), which describes the frequency behavior. Plancherel's theorem gives the possibility of extending ℱ to a unitary operator on L2(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
(1.4)Tyℱ-1(h)=ℱ-1(he-2πiy·),
where Ty is the translation operator.As a rule of thumb, smoothness off 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:
(1.5)ℋs2(ℝ):={f∈𝒮′(ℝ):∫ℝ‍|f^|2(1+|ω|2)sdω<∞}.
For s≥0, this is a subspace of L2(ℝ) consisting of smooth functions. For s<0 (negative smoothness), one has L2⊆ℋs2 and such spaces may include discrete measures and distributions. The respective inner product is of form
(1.6)〈f,h〉ℋs2=〈f^·v,h^·v〉,
where 〈·,·〉 is the L2-inner product and v=vso is as in (1.1).In terms of a generalized inner product, the inverse Fourier transform can be written asf(y)=〈f^,e-2πiy·〉. If we want the representation of f via (1.6), we have
(1.7)f(y)=〈f,Φy〉ℋs2=〈f^·v,ℱ(Φy)·v〉=〈f^,ℱ(Φy)·v2〉.
Thus ℱ(Φy)v2=e-2πiy·, so Φy=ℱ-1(v-2e-2πiy·). After applying (1.4), we derive the reproducing kernel for the Sobolev space ℋs2(ℝ) with respect to the classic inner product (1.6) to be
(1.8)Φy(t)=Φ(t,y)=Tyℱ-1(v-2)(t)
provided s>1/2 [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

Aweight is a positive function on ℝ2. By v, we denote a positive, even, submultiplicative weight such that v(0)=1, which satisfies the inequality v(z1+z2)≤v(z1)v(z2), for all z1, z2∈ℝ2. A positive, even weight function m on ℝ2 is called v-moderate if it satisfies for some C>0(1.9)m(z1+z2)≤Cv(z1)m(z2),∀z1,z2∈ℝ2.
Two weights m1 and m2 are called equivalent if one has for some C>0(1.10)C-1m1(z)≤m2(z)≤Cm1(z),foranyz∈ℝ2.
Commonly used weights are the (equivalent) submultiplicative weights of polynomial type (1+|ω|)s and (1+ω2)s/2, s≥0 because one has
(1.11)(1+ω2)s/2≤(1+|ω|)s≤2s/2(1+ω2)s/2.
It is easy to show that, if the moderate weight mo (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].


### 1.3. Short-Time Fourier Transform

By a time-frequency shift of a functionf, we mean MωTxf(t):=e2πit·ωf(t-x), for any pair (x,ω)∈ℝ2. 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):=MωTxg(y)¯, x,ω∈ℝ.Theshort-time Fourier transform (STFT for short notation) of a function f∈L2(ℝ) with respect to a windowL2-function g is
(1.12)Vgf(x,ω):=∫ℝ‍f(t)g(t-x)¯e-2πiω·tdt=〈f,MωTxg〉
for all (x,ω)∈ℝ2 and it holds ∥Vgf∥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 ofVgf 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 δy∈𝒟' by
(1.13)Vg(δy)(x,ω)=〈δy,MωTxg〉=MωTxg(y)¯,orVg(δy)=π*g(y).
The inversion formula for the STFT
(1.14)f=1〈γ,g〉∬‍Vgf(x,ω)MωTxγdxdω=1〈γ,g〉〈Vgf,π*γ〉
is well defined in the weak sense for all f∈L2(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 〈Vγ*F,h〉=〈F,Vγh〉 for F∈𝒮'(ℝ2) and h,γ∈𝒮(ℝ). Vγ* 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 g∈𝒮, a v-moderate weight m on ℝ2 and 1≤p, q<∞, the modulation spaceMmp,q(ℝ) consists of all tempered distributions f∈𝒮'(ℝ) for which Vgf has finite weighted mixed Lp,q-norm
(1.15)∥f∥Mmp,q:=(∫ℝ‍(∫ℝ‍|Vgf(x,ω)|pm(x,ω)pdx)q/pdω)1/q<∞
(with the usual adjustment for p, q=∞). If p=q, we write Mmp instead of Mmp,p and if m=1 then we write Mp,q instead of M1p,q. A special example is Feichtinger's algebra So=M1. It is sometimes useful to calculate Vgf with f∈So′ and g∈So, in particular when we want to go beyond L2-functions.Example 1.1.
Interesting examples of modulation spaces ([4], page 232) are(a)
Mm2(ℝ)=Lm2(ℝ) with m(x,ω)=m(x)(b)
in particularM2(ℝ)=L2(ℝ)(c)
Mm2(ℝ)=ℱLm2(ℝ) when m(x,ω)=m(ω)(d)
Mvso2(ℝ), which coincides with ℋs2(ℝ), s>1/2.Given any moderate weightm, the space Mm2(ℝ) is a time-frequency shift-invariant Hilbert space, with inner product
(1.16)〈f,h〉Mm2:=〈Vgf·m,Vgf·m〉L2(ℝ).
The norm deriving from this inner product is independent of the choice of the used window that is, two nonzero windows g and g0 produce equivalent norms, and there exists a constant c>0 such that for all f∈Mm2 it holds c-1∥Vgf∥Lm2≤∥Vg0f∥Lm2≤c∥Vgf∥Lm2. Similarly, equivalent weights m1 and m2 define the same modulation space.


## 1.1. Fourier Transform, Sobolev Space

Classical Fourier analysis employs two complementary representations to describe functions: the functionf itself (temporal behavior) and its Fourier transform
(1.3)ℱ(f)(ω)=f^(ω)=∫ℝ‍f(t)e-2πiωtdt
defined on L1(ℝ), which describes the frequency behavior. Plancherel's theorem gives the possibility of extending ℱ to a unitary operator on L2(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
(1.4)Tyℱ-1(h)=ℱ-1(he-2πiy·),
where Ty is the translation operator.As a rule of thumb, smoothness off 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:
(1.5)ℋs2(ℝ):={f∈𝒮′(ℝ):∫ℝ‍|f^|2(1+|ω|2)sdω<∞}.
For s≥0, this is a subspace of L2(ℝ) consisting of smooth functions. For s<0 (negative smoothness), one has L2⊆ℋs2 and such spaces may include discrete measures and distributions. The respective inner product is of form
(1.6)〈f,h〉ℋs2=〈f^·v,h^·v〉,
where 〈·,·〉 is the L2-inner product and v=vso is as in (1.1).In terms of a generalized inner product, the inverse Fourier transform can be written asf(y)=〈f^,e-2πiy·〉. If we want the representation of f via (1.6), we have
(1.7)f(y)=〈f,Φy〉ℋs2=〈f^·v,ℱ(Φy)·v〉=〈f^,ℱ(Φy)·v2〉.
Thus ℱ(Φy)v2=e-2πiy·, so Φy=ℱ-1(v-2e-2πiy·). After applying (1.4), we derive the reproducing kernel for the Sobolev space ℋs2(ℝ) with respect to the classic inner product (1.6) to be
(1.8)Φy(t)=Φ(t,y)=Tyℱ-1(v-2)(t)
provided s>1/2 [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

Aweight is a positive function on ℝ2. By v, we denote a positive, even, submultiplicative weight such that v(0)=1, which satisfies the inequality v(z1+z2)≤v(z1)v(z2), for all z1, z2∈ℝ2. A positive, even weight function m on ℝ2 is called v-moderate if it satisfies for some C>0(1.9)m(z1+z2)≤Cv(z1)m(z2),∀z1,z2∈ℝ2.
Two weights m1 and m2 are called equivalent if one has for some C>0(1.10)C-1m1(z)≤m2(z)≤Cm1(z),foranyz∈ℝ2.
Commonly used weights are the (equivalent) submultiplicative weights of polynomial type (1+|ω|)s and (1+ω2)s/2, s≥0 because one has
(1.11)(1+ω2)s/2≤(1+|ω|)s≤2s/2(1+ω2)s/2.
It is easy to show that, if the moderate weight mo (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].


## 1.3. Short-Time Fourier Transform

By a time-frequency shift of a functionf, we mean MωTxf(t):=e2πit·ωf(t-x), for any pair (x,ω)∈ℝ2. 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):=MωTxg(y)¯, x,ω∈ℝ.Theshort-time Fourier transform (STFT for short notation) of a function f∈L2(ℝ) with respect to a windowL2-function g is
(1.12)Vgf(x,ω):=∫ℝ‍f(t)g(t-x)¯e-2πiω·tdt=〈f,MωTxg〉
for all (x,ω)∈ℝ2 and it holds ∥Vgf∥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 ofVgf 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 δy∈𝒟' by
(1.13)Vg(δy)(x,ω)=〈δy,MωTxg〉=MωTxg(y)¯,orVg(δy)=π*g(y).
The inversion formula for the STFT
(1.14)f=1〈γ,g〉∬‍Vgf(x,ω)MωTxγdxdω=1〈γ,g〉〈Vgf,π*γ〉
is well defined in the weak sense for all f∈L2(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 〈Vγ*F,h〉=〈F,Vγh〉 for F∈𝒮'(ℝ2) and h,γ∈𝒮(ℝ). Vγ* 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 g∈𝒮, a v-moderate weight m on ℝ2 and 1≤p, q<∞, the modulation spaceMmp,q(ℝ) consists of all tempered distributions f∈𝒮'(ℝ) for which Vgf has finite weighted mixed Lp,q-norm
(1.15)∥f∥Mmp,q:=(∫ℝ‍(∫ℝ‍|Vgf(x,ω)|pm(x,ω)pdx)q/pdω)1/q<∞
(with the usual adjustment for p, q=∞). If p=q, we write Mmp instead of Mmp,p and if m=1 then we write Mp,q instead of M1p,q. A special example is Feichtinger's algebra So=M1. It is sometimes useful to calculate Vgf with f∈So′ and g∈So, in particular when we want to go beyond L2-functions.Example 1.1.
Interesting examples of modulation spaces ([4], page 232) are(a)
Mm2(ℝ)=Lm2(ℝ) with m(x,ω)=m(x)(b)
in particularM2(ℝ)=L2(ℝ)(c)
Mm2(ℝ)=ℱLm2(ℝ) when m(x,ω)=m(ω)(d)
Mvso2(ℝ), which coincides with ℋs2(ℝ), s>1/2.Given any moderate weightm, the space Mm2(ℝ) is a time-frequency shift-invariant Hilbert space, with inner product
(1.16)〈f,h〉Mm2:=〈Vgf·m,Vgf·m〉L2(ℝ).
The norm deriving from this inner product is independent of the choice of the used window that is, two nonzero windows g and g0 produce equivalent norms, and there exists a constant c>0 such that for all f∈Mm2 it holds c-1∥Vgf∥Lm2≤∥Vg0f∥Lm2≤c∥Vgf∥Lm2. Similarly, equivalent weights m1 and m2 define the same modulation space.


## 2. Reproducing Kernel with respect to av-Moderate Weight m

Given av-moderate weight m, we seek a two-dimensional function Φy(t):=Φ(t,y) such that for all f∈Mm2(ℝ), y∈ℝ, it holds f(y)=〈f,Φy〉Mm2(ℝ), that is
(2.1)f(y)=〈Vgf·m,VgΦy·m〉L2(ℝ2)=〈Vgf,m2·VgΦy〉L2(ℝ2).
We compare to (1.14) and conclude that π*g(y)=m2·VgΦy, that is,Theorem 2.1.
The reproducing kernel forMm2(ℝ) defined with respect to a v-moderate weight m is
(2.2)Φy=Vg*(m-2π*g(y)).
For y-fixed,
(2.3)∥Φy∥=∥m-1g(y-·)∥2.Proof.
Lety∈ℝ be fixed, we have
(2.4)∥Φy∥2=〈VgΦy·m,VgΦy·m〉=〈VgVg*(1m2·π*g(y))·m,VgVg*(1m2·π*g(y))·m〉=〈1m2·π*g(y)·m,1m2·π*g(y)·m〉=〈1m2·π*g(y),π*g(y)〉=∬‍1m(x,ω)2·MωTxg(y)MωTxg(y)¯dxdω=∬‍1m(x,ω)2|g(y-x)|2dxdω.Corollary 2.2.
Letf∈Mm12(ℝ) and let m2 be equivalent to m1 with equivalence constant c. Denote the respective reproducing kernels with Φ1 and Φ2.
Iff~(y)=〈f,Φ2,y〉Mm12 then
(2.5)c-2f(y)≤f~(y)≤c2f(y).Proof .
Instead of usingΦ1,y to reconstruct the function f at point y in the modulation space defined with respect to m1, we use m2 to obtain the approximation f~ as follows:
(2.6)f~(y)=〈Vgf·m1,VgΦ2,y·m1〉=〈Vgf·m1,m2-2π*g(y)·m1〉=〈Vgf,m2-2m12π*g(y)〉.
Then it holds c-2〈Vgf,π*g(y)〉≤f~(y)≤c2〈Vgf,π*g(y)〉, which is equivalent to (2.5).For instance, if the equivalence constant between weightsm1 and m2 is c=1.05, then for all f∈Mm12, it holds that 0.907f(y)≤f~(y)≤1.1025f(y).


### 2.1. Reproducing Kernel for the Sobolev Space

In analogy to (1.8), we calculate the RK for the modulation space Mv2(ℝ), equivalent to ℋv2(ℝ), defined with respect to weight (1.1). Since we now employ the inner product (2.1), the RK of Mv2(ℝ) depends on the used analysis window g.Let the RK function be denoted byΦy(t)=Φ(t,y), then (2.2) implies that Φy=Vg*(v-2π*g(y)), and by the pointwise version of (1.14)
(2.7)Φ(t,y)=∬‍1(1+ω2)se-2πiωyg(y-x)¯e2πiωtg(t-x)dxdω=∫‍e2πiω(t-y)(1+ω2)sdω∫‍g(y-x)¯g(t-x)dx.
We use (1.4) to obtain the RK formula (2.8).Corollary 2.3.
Lets>1/2 and v(x,ω)=(1+ω2)s/2. The reproducing kernel of Mv2(ℝ) is
(2.8)Φ(t,y)=Ty(ℱ-1(v-2)(t))·G(t,y),
where ∥Φy∥Mv22=2/(2s-1) and
(2.9)G(t,y)=∫‍g(t-x)g(y-x)¯dx=〈g,Tt-yg〉.Proof.
From (2.3), it follows
(2.10)∥Φy∥Mv22=∥v-1g(y-·)∥22=∬ℝ2‍1(1+|ω|2)s|g(y-x)|2dxdω=∫ℝ‍1(1+|ω|2)sdω·∫ℝ‍|g(y-x)|2dx=22s-1∥g∥22=22s-1.
The last equality indicates that the condition s>1/2 is necessary.Note 1.
G is decaying fast for a fast-decaying window g. For instance, if g(t)=e-πt2 we have
(2.11)G(t,y)=e-(π/2)(t-y)2∫‍e-2π(x-((t+y)/2))2dx=22e-(π/2)(t-y)2,
which decays fast the further away are t and y. In the discrete setting, G 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 G, which varies for different windows.


## 2.1. Reproducing Kernel for the Sobolev Space

In analogy to (1.8), we calculate the RK for the modulation space Mv2(ℝ), equivalent to ℋv2(ℝ), defined with respect to weight (1.1). Since we now employ the inner product (2.1), the RK of Mv2(ℝ) depends on the used analysis window g.Let the RK function be denoted byΦy(t)=Φ(t,y), then (2.2) implies that Φy=Vg*(v-2π*g(y)), and by the pointwise version of (1.14)
(2.7)Φ(t,y)=∬‍1(1+ω2)se-2πiωyg(y-x)¯e2πiωtg(t-x)dxdω=∫‍e2πiω(t-y)(1+ω2)sdω∫‍g(y-x)¯g(t-x)dx.
We use (1.4) to obtain the RK formula (2.8).Corollary 2.3.
Lets>1/2 and v(x,ω)=(1+ω2)s/2. The reproducing kernel of Mv2(ℝ) is
(2.8)Φ(t,y)=Ty(ℱ-1(v-2)(t))·G(t,y),
where ∥Φy∥Mv22=2/(2s-1) and
(2.9)G(t,y)=∫‍g(t-x)g(y-x)¯dx=〈g,Tt-yg〉.Proof.
From (2.3), it follows
(2.10)∥Φy∥Mv22=∥v-1g(y-·)∥22=∬ℝ2‍1(1+|ω|2)s|g(y-x)|2dxdω=∫ℝ‍1(1+|ω|2)sdω·∫ℝ‍|g(y-x)|2dx=22s-1∥g∥22=22s-1.
The last equality indicates that the condition s>1/2 is necessary.Note 1.
G is decaying fast for a fast-decaying window g. For instance, if g(t)=e-πt2 we have
(2.11)G(t,y)=e-(π/2)(t-y)2∫‍e-2π(x-((t+y)/2))2dx=22e-(π/2)(t-y)2,
which decays fast the further away are t and y. In the discrete setting, G 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 G, 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 functionb(x)≥0 describe the time-varying broadness of a strip STb in the time-frequency plane ℝ2=ℝ×ℝ^, that is,
(3.1)STb:={(x,ω)∈ℝ2:|ω|≤b(x)}.
We call the set STb a strip withvariable bandwidth(VB strip).The vertical distance functiondb at point z=(x,ω)∈ℝ2 is given by
(3.2)db(z):={0,ifz∈STb,|ω|-b(x),ifz∉STb.
In [5], we defined a variable bandwidth weight of order s on the time-frequency plane (VB weight) by
(3.3)mb,s(z)=(1+db(z))s,s>0
and proved it is a moderate weight. In this paper, we work with a Sobolev-type weight
(3.4)mb,s(z)=(1+db(z)2)s/2,s>0,
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. 						(3.5)ma(z)={1,if|ω|≤a,(1+(|ω|-a)2)s/2,otherwise.Notice that, whenever we choosea≡0 in the last equation, we get the standard Sobolev weight (1+|ω|2)s/2.We choose to work withs>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 STb. It can be shown that, for a well-localizing window g, if the weighted Vgf is integrable for some s>0, then Vgf 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 Proposition 3.1 from [5], the following result holds.Proposition 3.1.
Lets>0, take b to be a nonnegative function on ℝ and define weight (3.4) for db given as in (3.2). If for some k≥0 the boundary function b satisfies
(3.6)(∀x,y∈ℝ)|b(x)-b(y)|≤k|x-y|,
then the weight mb is moderate with respect to
(3.7)vs(x,ω)=(1+(|x|+|ω|)2)s/2,s>0,
That is, it holds mb(x1+x2,ω1+ω2)≤ksvs(x1,ω1)mb(x2,ω2) for all x1, x2, ω1, ω2∈ℝ.As a consequence,mb 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.Corollary 3.2.
Ifb is a bounded function, then the weight mb, given by (3.4), is moderate with respect to vso(x,ω)=(1+|ω|2)s/2.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|h(x)|<c for all x∈ℝ, and let b≥0 generate a moderate weight mb (3.4). Then b+h generates a moderate weight mb+h, equivalent to mb, provided b+h≥0.Proof.
We work here withc=2/2: let 0≤h(x)<2/2 for all x∈ℝ. The equivalence inequality is trivially satisfied on STb+h as both weights have value 1. Let b(x)<|ω|≤b(x)+h(x). Then |ω|-b(x)≤h(x), mb+h(x,ω)=1 and we have
(3.8)mb+h(x,ω)=1<mb(x,ω)≤(1+h(x)2)s/2≤2s/2·1=2s/2mb+h(x,ω).
If |ω|-m(x)-h(x)>0, then both weights have nontrivial values. Using this estimate
(3.9)0≤(|ω|-b(x)-2h(x))2+1-2h2(x),
which holds true whenever h(x)<2/2, we derive that
(3.10)1+(|ω|-b(x))2≤2(1+(|ω|-b(x)-h(x))2).
Then  it  holdsmb+h(x,ω)≤mb(x,ω)≤2s/2mb+h(x,ω).Equivalent weights are moderate simultaneously, thusmb+h is moderate.If we apply Proposition3.3 a finite number for shifting the constant bandwidth in (3.5) toward 0, we have the following Corollary.Corollary 3.4.
A VB weightmb, defined with respect to a bounded bandwidth b, is equivalent to (1+|ω|2)s/2.


### 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.
Letf be defined on ℝ and fix s>0. We say that f is a function with variable bandwidth b=b(t) if it belongs to a modulation space Mmp,q(ℝ) for some 1≤p,q≤∞ and a VB weight m=mb,s, related to b and defined as in (3.4).We will call the spaceMmp,q(ℝ) a variable bandwidth space and refer to it by VB space when parameters p,q are known and m=mb,s is fixed up to equivalence. As usual, if p=q we write Mmp instead of Mmp,p. The modulation space (and VB space) norm in this paper is marked with ∥·∥ or ∥·∥Mmp,q, 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].


## 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.
Letf be defined on ℝ and fix s>0. We say that f is a function with variable bandwidth b=b(t) if it belongs to a modulation space Mmp,q(ℝ) for some 1≤p,q≤∞ and a VB weight m=mb,s, related to b and defined as in (3.4).We will call the spaceMmp,q(ℝ) a variable bandwidth space and refer to it by VB space when parameters p,q are known and m=mb,s is fixed up to equivalence. As usual, if p=q we write Mmp instead of Mmp,p. The modulation space (and VB space) norm in this paper is marked with ∥·∥ or ∥·∥Mmp,q, 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-spaceMmb2(ℝ), defined with respect to (3.4), is Φy=Vg*(mb-2·π*g(y)) and the norm of Φy depends on the bandwidth function b(4.1)∥Φy∥2=2∥g(y-.)b(·)∥22+22s-1,s>12.Proof.
Due to (2.3) and the symmetry properties of mb (3.4), it holds
(4.2)∥Φy∥2=∬‍1m(x,ω)2|g(y-x)|2dxdω=∫‍|g(y-x)|2(∫-b(x)b(x)‍1dω+2∫b(x)+∞‍1(1+ω-b(x))2sdω)dx=2∫‍|g(y-x)|2(b(x)+∫1+∞‍1ξ2sdξ)dx=2∫‍|g(y-x)|2(b(x)+12s-1)dx=2∥g(y-.)b(·)∥22+22s-1∥g∥22.
Our a priori assumption is that we work with a normalized window g that is, ∥g∥2=1, so (4.1) holds.A similar split of integration, as seen in the proof of Corollary2.3, is possible for a constant bandwidth weight ma (3.5). Locally, ma depends only on ω.Corollary 4.2.
The RK forMma2(ℝ), where ma is as in (3.5), is
(4.3)Φ(t,y)=G(t,y)e2πia(t-y)-e-2πia(t-y)2πi(t-y)+G(t,y)(e-2πia(t-y)Tyℱ((vso)+-2)(t)+e2πia(t-y)Tyℱ-1((vso)+-2)(t)),
where G is defined in (2.9).Proof.
By (2.2), we have Φy(t)=∫g(t-x)g(y-x)¯(∫(e2πiω(t-y)/ma2(x,ω))dω)dx. Recall (3.5) and its symmetry property; therefore, we can use the notation (vso)+ to describe the weight (vso(x,ω))+=(1+ω2)s/2, for x>0 and (vso(x,ω))+=0 otherwise and obtain
(4.4)∫e2πiω(t-y)ma2(x,ω)dω=∫-aa‍e2πiω(t-y)dω+∫a+∞‍e2πiω(t-y)(1+(ω-a)2)-sdω+∫-∞-a‍e2πiω(t-y)(1+(-ω-a)2)-sdω=e2πia(t-y)-e-2πia(t-y)2πi(t-y)+e2πia(t-y)∫0∞‍e2πiξ(t-y)(1+ξ2)-sdξ+e-2πia(t-y)∫0∞‍e-2πiξ(t-y)(1+ξ2)-sdξ=e2πia(t-y)-e-2πia(t-y)2πi(t-y)+e-2πia(t-y)Tyℱ((vso)+-2)(t)+e2πia(t-y)Tyℱ-1((vso)+-2)(t).



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*Source: 469341-2012-09-25.xml*