Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators
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Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators Zhongyan Li1
· Deguang Han2
Received: 12 October 2017 / Accepted: 2 June 2018 © Springer Nature B.V. 2018
Abstract We consider the problem of characterizing the bounded linear operator multipliers on L2 (R) that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix M(t) = [fij (t)]m×m (where fij ∈ L∞ (R)) is a multiplier for Parseval Gabor multi-frame generators with parameters a, b > 0 if and only if M(t) is unitary and M ∗ (t)M(t + b1 ) = λ(t)I for some unimodular a-periodic function λ(t). As a special case (m = 1) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators. Keywords Gabor family · Gabor multi-frame generators · Functional matrix multipliers Mathematics Subject Classification (2010) 42C15 · 46C05 · 47B10
1 Introduction Motivated by the characterization of functional wavelet multipliers in the Wutam Consortium paper [11], Gu and Han [8] investigated the functional Gabor frame multipliers and obtained the following characterization: a L∞ -function h on R is a functional Gabor frame multiplier for the time-frequency lattice aZ × bZ if and only if it is unimodular and h(t)/ h(t + 1/b) is a-periodic, where by a functional Gabor frame multiplier h we mean that Zhongyan Li acknowledges the support from the National Natural Science Foundation of China (Grant No. 11571107); Deguang Han acknowledges the support from NSF under the grants DMS-1403400 and DMS-1712602.
B Z. Li
[email protected] D. Han [email protected]
1
Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China
2
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
Z. Li, D. Han
hg is a Parseval Gabor frame generator for L2 (R) whenever g is a Parseval Gabor frame generator for L2 (R). It is natural to ask if there is a similar characterization for Gabor frames with multi-generators in view of many studies of Gabor multi-frames (cf. [3, 4, 6, 7, 9, 10] and the references therein). The purpose of this note is to present such a generalization. Given a, b > 0 and g ∈ L2 (R). The Gabor (or Weyl-Heisenberg) family (g, a, b) is defined to be the collection of functions: 2π ikbt g(t − na) : k, n ∈ Z . e Definition 1.1 Let g1 , . . . , gL ∈ L2 (R). We say that G(t) = (g1 (t), . . . , gL (t))τ is a Gabor multi-frame generator of length L if {(gj , a, b), j = 1, 2, . . . , L} is a frame for L2 (R), i.e., there exist A, B > 0, such that Af 2 ≤
L 2π ikbt 2 f, e gj (t − na) ≤ Bf 2 i=1 k,n∈Z
holds for all f ∈ L2 (R). It is called a Parseval Gabor multi-frame generator if A = B = 1. It is well-known that a single function Gabor frame generator exists if and only if ab ≤ 1 (cf. [2]). For the case L−1 < ab ≤ L, we need at least L functions to generate a Gabor frame for L2 (R) (cf. [5]). Therefore if there exists a Gabor multi-frame generator (g1 , . . . , gm ) for L2 (R) and L − 1 < ab ≤
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