On the frame set of the second-order B-spline

https://doi.org/10.1016/j.acha.2022.08.007Get rights and content

Abstract

The frame set of a function g∈L2(R) is the set of all parameters (a,b)∈R+2 for which the collection of time-frequency shifts of g along aZ×bZ form a Gabor frame for L2(R). Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second-order B-spline. Our method uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.

Section snippets

Introduction and main results

Given a window function g∈L2(R) and constants a,b>0, the collection of time-frequency shiftsG(g,a,b)={MℓbTkag=e2πiℓb⋅g(⋅−ka):(ℓ,k)∈Z2} is called a Gabor frame for L2(R) if there exist constants A,B>0 such thatA‖f‖22≤∑ℓ,k∈Z|〈f,MℓbTkag〉|2≤B‖f‖22, for all f∈L2(R). When G(g,a,b) is a Gabor frame, there exists a dual window γ∈L2(R) such that G(γ,a,b) is also a Gabor frame for L2(R) called the (canonical) dual to G(g,a,b). Consequently, for any f∈L2(R) we have the following reconstruction formulas:f=∑

F(B2) contains Λ

In this section we prove the first part of Theorem 1 by establishing the following result.

Theorem 2

For m≥3, let (a,b)∈Λm. Then, the Gabor system G(B2,a,b) is a frame for L2(R), and there is a unique dual window h∈L2(R) such that supp(h)⊆[−2m−12a,2m−12a]. Furthermore, for each (a,b)∈Λ, the Gabor system G(B2,a,b) is a frame for L2(R).

To prove Theorem 2 we only need to show that (3) has a solution h∈L2(R) that is a bounded and compactly supported function. As mentioned earlier, the determinant of the

F(B2) contains Γ3

We now turn to the second part of Theorem 1 by showing that the following result holds.

Theorem 3

Let (a,b)∈Γ3. Then, the Gabor system G(B2,a,b) is a frame for L2(R), and there is a unique dual window h∈L2(R) such that supp(h)⊆[−5a2,5a2].

We observe that when (a,b)∈Γ3, the matrix G3 becomesG3=(Dv0g2,2), where 0 is a 1×4 matrix of 0s, v is a column vector in R4 and D denotes the 4×4 matrix obtained by deleting the last row and the last column of G3 and given byD=(g−2,−2g−2,−100g−1,−2g−1,−1g−1,0g−1,1g0,−2g0,−

Acknowledgement

Part of this work was completed while the first-named author was a visiting graduate student in the Department of Mathematics at the University of Maryland during the Fall 2017 semester. He would like to thank the Department for its hospitality and the African Center of Excellence in Mathematics and Application (CEA-SMA) at the Institut de Mathématiques et de Sciences Physiques (IMSP) for funding his visit. K.A. Okoudjou was partially supported by a grant from the Simons Foundation # 319197, by

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    Yébéni B. Kouagou suddenly passed away in 2018, a few weeks after the first version of this work was released. This version is dedicated to his memory.

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