Reproducing Kernels and Variable Bandwidth

Abstract

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 if 𝑀²_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.

1. Motivation and Preliminaries

Classical Sobolev spaces ℋ²_𝑠(ℝ) [1, 2] are defined by weighting on the frequency side with 𝑣^𝑜_𝑠(𝜔)=1+𝜔^2𝑠/2. (1.1) It is submultiplicative for 𝑠≥0 and defines a reproducing kernel Hilbert space (RKHS), whenever 𝑠>1/2 [3]. Knowing that modulation spaces 𝑀²_𝑣𝑜𝑠(ℝ), defined with respect to (1.1), coincide with ℋ²_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=𝑀²_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 property𝑓𝑦=𝑓,Φ_𝑦ℳ (1.2) 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.

1.1. Fourier Transform, Sobolev Space

Classical Fourier analysis employs two complementary representations to describe functions: the function 𝑓 itself (temporal behavior) and its Fourier transform ℱ𝑓(𝜔)=𝑓(𝜔)=_ℝ𝑓(𝑡)𝑒^{−2𝜋𝑖𝜔𝑡}𝑑𝑡 (1.3) 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 𝑇_𝑦ℱ⁻¹(ℎ)=ℱ⁻¹ℎ𝑒^{−2𝜋𝑖𝑦⋅}, (1.4) 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: ℋ²_𝑠(ℝ)∶=𝑓∈𝒮^′(ℝ)∶_ℝ𝑓²1+|𝜔|^2𝑠𝑑𝜔<∞. (1.5) For 𝑠≥0, this is a subspace of 𝐿²(ℝ) consisting of smooth functions. For 𝑠<0 (negative smoothness), one has 𝐿²⊆ℋ²_𝑠 and such spaces may include discrete measures and distributions. The respective inner product is of form 𝑓,ℎ_ℋ2𝑠=𝑓⋅𝑣,ℎ⋅𝑣, (1.6) where ⟨⋅,⋅⟩ is the 𝐿²-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 𝑓𝑦=𝑓,Φ_𝑦ℋ2𝑠=⟨𝑓⋅𝑣,ℱΦ_𝑦⋅𝑣⟩=𝑓,ℱΦ_𝑦⋅𝑣². (1.7) Thus ℱ(Φ_𝑦)𝑣²=𝑒^{−2𝜋𝑖𝑦⋅}, so Φ_𝑦=ℱ⁻¹(𝑣⁻²𝑒^{−2𝜋𝑖𝑦⋅}). After applying (1.4), we derive the reproducing kernel for the Sobolev space ℋ²_𝑠(ℝ) with respect to the classic inner product (1.6) to be Φ_𝑦(𝑡)=Φ𝑡,𝑦=𝑇_𝑦ℱ⁻¹𝑣⁻²(𝑡) (1.8) 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

A weight 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>0𝑚𝑧₁+𝑧₂≤𝐶𝑣𝑧₁𝑚𝑧₂,∀𝑧₁,𝑧₂∈ℝ². (1.9) Two weights 𝑚₁ and 𝑚₂ are called equivalent if one has for some 𝐶>0𝐶⁻¹𝑚₁(𝑧)≤𝑚₂(𝑧)≤𝐶𝑚₁(𝑧),forany𝑧∈ℝ². (1.10) 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+𝜔^2𝑠/2≤(1+|𝜔|)^𝑠≤2^𝑠/21+𝜔^2𝑠/2. (1.11) 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].

1.3. Short-Time Fourier Transform

By a time-frequency shift of a function 𝑓, we mean 𝑀_𝜔𝑇_𝑥𝑓(𝑡)∶=𝑒^{2𝜋𝑖𝑡⋅𝜔}𝑓(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)∶=𝑀_𝜔𝑇_𝑥𝑔(𝑦), 𝑥,𝜔∈ℝ.

The short-time Fourier transform (STFT for short notation) of a function 𝑓∈𝐿²(ℝ) with respect to a window 𝐿²-function 𝑔 is 𝑉_𝑔𝑓(𝑥,𝜔)∶=_ℝ𝑓(𝑡)𝑔(𝑡−𝑥)𝑒^{−2𝜋𝑖𝜔⋅𝑡}𝑑𝑡=𝑓,𝑀_𝜔𝑇_𝑥𝑔 (1.12) for all (𝑥,𝜔)∈ℝ² and it holds ‖𝑉_𝑔𝑓‖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_𝑔𝑓 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 𝑉_𝑔𝛿_𝑦(𝑥,𝜔)=𝛿_𝑦,𝑀_𝜔𝑇_𝑥𝑔=𝑀_𝜔𝑇_𝑥𝑔𝑦,or𝑉_𝑔𝛿_𝑦=𝜋^∗𝑔𝑦. (1.13) The inversion formula for the STFT 1𝑓=𝑉𝛾,𝑔^𝑔𝑓(𝑥,𝜔)𝑀_𝜔𝑇_𝑥1𝛾𝑑𝑥𝑑𝜔=𝑉𝛾,𝑔^𝑔𝑓,𝜋^∗𝛾 (1.14) 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 ⟨𝑉^∗_𝛾𝐹,ℎ⟩=⟨𝐹,𝑉_𝛾ℎ⟩ for 𝐹∈𝒮′(ℝ²) and ℎ,𝛾∈𝒮(ℝ). 𝑉^∗_𝛾 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 space 𝑀_𝑚^𝑝,𝑞(ℝ) consists of all tempered distributions 𝑓∈𝒮′(ℝ) for which 𝑉_𝑔𝑓 has finite weighted mixed 𝐿^𝑝,𝑞-norm 𝑓_𝑀𝑚^𝑝,𝑞∶=_ℝℝ𝑉_𝑔𝑓(𝑥,𝜔)^𝑝𝑚(𝑥,𝜔)^𝑝𝑑𝑥^𝑞/𝑝𝑑𝜔^1/𝑞<∞ (1.15) (with the usual adjustment for 𝑝, 𝑞=∞). If 𝑝=𝑞, we write 𝑀^𝑝_𝑚 instead of 𝑀_𝑚^𝑝,𝑝 and if 𝑚=1 then we write 𝑀^𝑝,𝑞 instead of 𝑀₁^𝑝,𝑞. A special example is Feichtinger's algebra 𝑆_𝑜=𝑀¹. It is sometimes useful to calculate 𝑉_𝑔𝑓 with 𝑓∈𝑆^′_𝑜 and 𝑔∈𝑆_𝑜, in particular when we want to go beyond 𝐿²-functions.

Example 1.1. Interesting examples of modulation spaces ([4], page 232) are(a)𝑀²_𝑚(ℝ)=𝐿²_𝑚(ℝ) with 𝑚(𝑥,𝜔)=𝑚(𝑥)(b)in particular 𝑀²(ℝ)=𝐿²(ℝ)(c)𝑀²_𝑚(ℝ)=ℱ𝐿²_𝑚(ℝ) when 𝑚(𝑥,𝜔)=𝑚(𝜔)(d)𝑀²_𝑣𝑜𝑠(ℝ), which coincides with ℋ²_𝑠(ℝ), 𝑠>1/2.

Given any moderate weight 𝑚, the space 𝑀²_𝑚(ℝ) is a time-frequency shift-invariant Hilbert space, with inner product 𝑓,ℎ_𝑀2𝑚∶=𝑉_𝑔𝑓⋅𝑚,𝑉_𝑔𝑓⋅𝑚_𝐿2(ℝ). (1.16) 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∈𝑀²_𝑚 it holds 𝑐⁻¹‖𝑉_𝑔𝑓‖_𝐿2𝑚≤‖𝑉_𝑔0𝑓‖_𝐿2𝑚≤𝑐‖𝑉_𝑔𝑓‖_𝐿2𝑚. Similarly, equivalent weights 𝑚₁ and 𝑚₂ define the same modulation space.

2. Reproducing Kernel with respect to a 𝑣-Moderate Weight 𝑚

Given a 𝑣-moderate weight 𝑚, we seek a two-dimensional function Φ_𝑦(𝑡)∶=Φ(𝑡,𝑦) such that for all 𝑓∈𝑀²_𝑚(ℝ), 𝑦∈ℝ, it holds 𝑓(𝑦)=⟨𝑓,Φ_𝑦⟩_{𝑀2𝑚(ℝ)}, that is 𝑓𝑦=𝑉_𝑔𝑓⋅𝑚,𝑉_𝑔Φ_𝑦⋅𝑚_𝐿2(ℝ2)=𝑉_𝑔𝑓,𝑚²⋅𝑉_𝑔Φ_{𝑦𝐿2(ℝ2)}. (2.1) We compare to (1.14) and conclude that 𝜋^∗𝑔(𝑦)=𝑚²⋅𝑉_𝑔Φ_𝑦, that is,

Theorem 2.1. The reproducing kernel for 𝑀²_𝑚(ℝ) defined with respect to a 𝑣-moderate weight 𝑚 is Φ_𝑦=𝑉^∗_𝑔𝑚⁻²𝜋^∗𝑔𝑦. (2.2) For 𝑦-fixed, Φ_𝑦=𝑚⁻¹𝑔𝑦−⋅₂. (2.3)

Proof. Let 𝑦∈ℝ be fixed, we have Φ_𝑦²=𝑉_𝑔Φ_𝑦⋅𝑚,𝑉_𝑔Φ_𝑦=𝑉⋅𝑚_𝑔𝑉^∗_𝑔1𝑚²⋅𝜋^∗𝑔𝑦⋅𝑚,𝑉_𝑔𝑉^∗_𝑔1𝑚²⋅𝜋^∗𝑔𝑦=1⋅𝑚𝑚²⋅𝜋^∗𝑔𝑦1⋅𝑚,𝑚²⋅𝜋^∗𝑔𝑦=1⋅𝑚𝑚²⋅𝜋^∗𝑔𝑦,𝜋^∗𝑔𝑦=1𝑚(𝑥,𝜔)²⋅𝑀_𝜔𝑇_𝑥𝑔𝑦𝑀_𝜔𝑇_𝑥𝑔𝑦1𝑑𝑥𝑑𝜔=𝑚(𝑥,𝜔)²𝑔𝑦−𝑥²𝑑𝑥𝑑𝜔. (2.4)

Corollary 2.2. Let 𝑓∈𝑀²_𝑚1(ℝ) and let 𝑚₂ be equivalent to 𝑚₁ with equivalence constant 𝑐. Denote the respective reproducing kernels with Φ₁ and Φ₂.
If 𝑓(𝑦)=⟨𝑓,Φ_2,𝑦⟩_𝑀2𝑚1 then 𝑐⁻²𝑓𝑦≤𝑓𝑦≤𝑐²𝑓𝑦. (2.5)

Proof . Instead of using Φ_1,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: 𝑓𝑦=𝑉_𝑔𝑓⋅𝑚₁,𝑉_𝑔Φ_2,𝑦⋅𝑚₁=𝑉_𝑔𝑓⋅𝑚₁,𝑚₂⁻²𝜋^∗𝑔𝑦⋅𝑚₁=𝑉_𝑔𝑓,𝑚₂⁻²𝑚²₁𝜋^∗𝑔𝑦. (2.6) Then it holds 𝑐⁻²⟨𝑉_𝑔𝑓,𝜋^∗𝑔(𝑦)⟩≤𝑓(𝑦)≤𝑐²⟨𝑉_𝑔𝑓,𝜋^∗𝑔(𝑦)⟩, which is equivalent to (2.5).

For instance, if the equivalence constant between weights 𝑚₁ and 𝑚₂ is 𝑐=1.05, then for all 𝑓∈𝑀²_𝑚1, 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 𝑀²_𝑣(ℝ), equivalent to ℋ²_𝑣(ℝ), defined with respect to weight (1.1). Since we now employ the inner product (2.1), the RK of 𝑀²_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⁻²𝜋^∗𝑔(𝑦)), and by the pointwise version of (1.14) Φ=1𝑡,𝑦1+𝜔^2𝑠𝑒^{−2𝜋𝑖𝜔𝑦}𝑔𝑦−𝑥𝑒^{2𝜋𝑖𝜔𝑡}𝑔()𝑒𝑡−𝑥𝑑𝑥𝑑𝜔=^{2𝜋𝑖𝜔(𝑡−𝑦)}1+𝜔^2𝑠𝑑𝜔𝑔𝑦−𝑥𝑔(𝑡−𝑥)𝑑𝑥. (2.7) We use (1.4) to obtain the RK formula (2.8).

Corollary 2.3. Let 𝑠>1/2 and 𝑣(𝑥,𝜔)=(1+𝜔²)^𝑠/2. The reproducing kernel of 𝑀²_𝑣(ℝ) is Φ𝑡,𝑦=𝑇_𝑦ℱ⁻¹𝑣⁻²(𝑡)⋅𝐺𝑡,𝑦, (2.8) where ‖Φ_𝑦‖²_𝑀2𝑣=2/(2𝑠−1) and 𝐺𝑡,𝑦=𝑔(𝑡−𝑥)𝑔𝑦−𝑥𝑑𝑥=𝑔,𝑇_𝑡−𝑦𝑔. (2.9)

Proof. From (2.3), it follows Φ_𝑦²_𝑀2𝑣=𝑣⁻¹𝑔𝑦−⋅²₂=_ℝ211+|𝜔|^2𝑠𝑔𝑦−𝑥²=𝑑𝑥𝑑𝜔_ℝ11+|𝜔|^2𝑠𝑑𝜔⋅_ℝ𝑔𝑦−𝑥²2𝑑𝑥=𝑔2𝑠−1²²=2.2𝑠−1 (2.10) 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 𝐺𝑡,𝑦=𝑒^{−(𝜋/2)(𝑡−𝑦)2}𝑒^{−2𝜋(𝑥−((𝑡+𝑦)/2))2}√𝑑𝑥=22𝑒^{−(𝜋/2)(𝑡−𝑦)2}, (2.11) 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 function 𝑏(𝑥)≥0 describe the time-varying broadness of a strip ST_𝑏 in the time-frequency plane ℝ²=ℝ×ℝ, that is, ST_𝑏∶=(𝑥,𝜔)∈ℝ²∶|𝜔|≤𝑏(𝑥). (3.1) We call the set ST_𝑏 a strip with variable bandwidth (VB strip).

The vertical distance function 𝑑_𝑏 at point 𝑧=(𝑥,𝜔)∈ℝ² is given by 𝑑_𝑏(𝑧)∶=0,if𝑧∈ST_𝑏,|𝜔|−𝑏(𝑥),if𝑧∉ST_𝑏. (3.2) In [5], we defined a variable bandwidth weight of order 𝑠 on the time-frequency plane (VB weight) by 𝑚_𝑏,𝑠(𝑧)=1+𝑑_𝑏(𝑧)^𝑠,𝑠>0 (3.3) and proved it is a moderate weight. In this paper, we work with a Sobolev-type weight 𝑚_𝑏,𝑠(𝑧)=1+𝑑_𝑏(𝑧)^2𝑠/2,𝑠>0, (3.4) 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. 𝑚_𝑎(𝑧)=|𝜔|𝜔|)1,if≤𝑎,1+(|−𝑎^2𝑠/2,otherwise. (3.5)

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_𝑔𝑓 is integrable for some 𝑠>0, then 𝑉_𝑔𝑓 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. 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∈ℝ𝑏(𝑥)−𝑏𝑦≤𝑘𝑥−𝑦, (3.6) then the weight 𝑚_𝑏 is moderate with respect to 𝑣_𝑠(𝑥,𝜔)=1+(|𝑥|+|𝜔|)^2𝑠/2,𝑠>0, (3.7) That is, it holds 𝑚_𝑏(𝑥₁+𝑥₂,𝜔₁+𝜔₂)≤𝑘^𝑠𝑣_𝑠(𝑥₁,𝜔₁)𝑚_𝑏(𝑥₂,𝜔₂) for all 𝑥₁, 𝑥₂, 𝜔₁, 𝜔₂∈ℝ.

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.

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.

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 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 𝑚_𝑏+ℎ(𝑥,𝜔)=1<𝑚_𝑏(𝑥,𝜔)≤1+ℎ(𝑥)^2𝑠/2≤2^𝑠/2⋅1=2^𝑠/2𝑚_𝑏+ℎ(𝑥,𝜔). (3.8) If |𝜔|−𝑚(𝑥)−ℎ(𝑥)>0, then both weights have nontrivial values. Using this estimate 0≤(|𝜔|−𝑏(𝑥)−2ℎ(𝑥))²+1−2ℎ²(𝑥), (3.9) which holds true whenever ℎ(𝑥)<√2/2, we derive that 1+(|𝜔|−𝑏(𝑥))²≤21+(|𝜔|−𝑏(𝑥)−ℎ(𝑥))². (3.10)
Then it holds 𝑚_𝑏+ℎ(𝑥,𝜔)≤𝑚_𝑏(𝑥,𝜔)≤2^𝑠/2𝑚_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.

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 𝑓 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 𝑀_𝑚^𝑝,𝑞(ℝ) for some 1≤𝑝,𝑞≤∞ and a VB weight 𝑚=𝑚_𝑏,𝑠, related to 𝑏 and defined as in (3.4).

We will call the space 𝑀_𝑚^𝑝,𝑞(ℝ) a variable bandwidth space and refer to it by VB space when parameters 𝑝,𝑞 are known and 𝑚=𝑚_𝑏,𝑠 is fixed up to equivalence. As usual, if 𝑝=𝑞 we write 𝑀^𝑝_𝑚 instead of 𝑀_𝑚^𝑝,𝑝. The modulation space (and VB space) norm in this paper is marked with ‖⋅‖ or ‖⋅‖_𝑀𝑚^𝑝,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-space 𝑀²_𝑚𝑏(ℝ), defined with respect to (3.4), is Φ_𝑦=𝑉^∗_𝑔(𝑚_𝑏⁻²⋅𝜋^∗𝑔(𝑦)) and the norm of Φ_𝑦 depends on the bandwidth function 𝑏Φ_𝑦²=2𝑔𝑦−.𝑏(⋅)²₂+212𝑠−1,𝑠>2. (4.1)

Proof. Due to (2.3) and the symmetry properties of 𝑚_𝑏 (3.4), it holds Φ_𝑦²=1𝑚(𝑥,𝜔)²𝑔𝑦−𝑥²=𝑔𝑑𝑥𝑑𝜔𝑦−𝑥²_(𝑥)^𝑏(𝑥)_−𝑏1𝑑𝜔+2_𝑏(𝑥)^+∞1(1+𝜔−𝑏(𝑥))^2𝑠𝑔𝑑𝜔𝑑𝑥=2𝑦−𝑥²𝑏(𝑥)+₁^+∞1𝜉^2𝑠𝑔𝑑𝜉𝑑𝑥=2𝑦−𝑥²𝑏(𝑥)+1𝑔2𝑠−1𝑑𝑥=2𝑦−.𝑏(⋅)²₂+2𝑔2𝑠−1²². (4.2) 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 Corollary 2.3, is possible for a constant bandwidth weight 𝑚_𝑎 (3.5). Locally, 𝑚_𝑎 depends only on 𝜔.

Corollary 4.2. The RK for 𝑀²_𝑚𝑎(ℝ), where 𝑚_𝑎 is as in (3.5), is Φ𝑒𝑡,𝑦=𝐺𝑡,𝑦^{2𝜋𝑖𝑎(𝑡−𝑦)}−𝑒^{−2𝜋𝑖𝑎(𝑡−𝑦)}𝑒2𝜋𝑖𝑡−𝑦+𝐺𝑡,𝑦^{−2𝜋𝑖𝑎(𝑡−𝑦)}𝑇_𝑦ℱ𝑣^𝑜_𝑠+⁻²(𝑡)+𝑒^{2𝜋𝑖𝑎(𝑡−𝑦)}𝑇_𝑦ℱ⁻¹𝑣^𝑜_𝑠+⁻²(𝑡), (4.3) where 𝐺 is defined in (2.9).

Proof. By (2.2), we have Φ_𝑦(𝑡)=∫𝑔(𝑡−𝑥)𝑔(𝑦−𝑥)(∫(𝑒^{2𝜋𝑖𝜔(𝑡−𝑦)}/𝑚²_𝑎(𝑥,𝜔))𝑑𝜔)𝑑𝑥. Recall (3.5) and its symmetry property; therefore, we can use the notation (𝑣^𝑜_𝑠)₊ to describe the weight (𝑣^𝑜_𝑠(𝑥,𝜔))₊=(1+𝜔²)^𝑠/2, for 𝑥>0 and (𝑣^𝑜_𝑠(𝑥,𝜔))₊=0 otherwise and obtain 𝑒^{2𝜋𝑖𝜔(𝑡−𝑦)}𝑚²_𝑎(𝑥,𝜔)𝑑𝜔=^𝑎_−𝑎𝑒^{2𝜋𝑖𝜔(𝑡−𝑦)}𝑑𝜔+_𝑎^+∞𝑒^{2𝜋𝑖𝜔(𝑡−𝑦)}1+(𝜔−𝑎)^2−𝑠+𝑑𝜔_{−𝑎−∞}𝑒^{2𝜋𝑖𝜔(𝑡−𝑦)}1+(−𝜔−𝑎)^2−𝑠=𝑒𝑑𝜔_{2𝜋𝑖𝑎(𝑡−𝑦)}−𝑒^{−2𝜋𝑖𝑎(𝑡−𝑦)}2𝜋𝑖𝑡−𝑦+𝑒^{2𝜋𝑖𝑎(𝑡−𝑦)∞}₀𝑒^{2𝜋𝑖𝜉(𝑡−𝑦)}1+𝜉^2−𝑠𝑑𝜉+𝑒^{−2𝜋𝑖𝑎(𝑡−𝑦)∞}₀𝑒^{−2𝜋𝑖𝜉(𝑡−𝑦)}1+𝜉^2−𝑠=𝑒𝑑𝜉_{2𝜋𝑖𝑎(𝑡−𝑦)}−𝑒^{−2𝜋𝑖𝑎(𝑡−𝑦)}2𝜋𝑖𝑡−𝑦+𝑒^{−2𝜋𝑖𝑎(𝑡−𝑦)}𝑇_𝑦ℱ𝑣^𝑜_𝑠+⁻²(𝑡)+𝑒^{2𝜋𝑖𝑎(𝑡−𝑦)}𝑇_𝑦ℱ⁻¹𝑣^𝑜_𝑠+⁻²(𝑡). (4.4)

Acknowledgment

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