Subspace mixed rational time-frequency multiwindow Gabor frames and their Gabor duals

Yan Zhang,Yun-Zhang Li

doi:10.1186/s13660-018-1876-7

Abstract

For a usual multiwindow Gabor system, all windows share common time-frequency shifts. A mixed multiwindow Gabor system is one of its generalizations, for which time-frequency shifts vary with the windows. This paper addresses subspace mixed multiwindow Gabor systems with rational time-frequency product lattices. It is a continuation of (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013; Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014). In (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013) we dealt with discrete subspace mixed Gabor systems and in (Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014) with L^{2}(mathbb{R}) ones. In this paper, using a suitable Zak transform matrix method, we characterize subspace mixed multiwindow Gabor frames and their Gabor duals, obtain explicit expressions of Gabor duals, and characterize the uniqueness of Gabor duals. We also provide some examples, which show that there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames.

Highlights

Let H be a separable Hilbert space
4 Conclusions A mixed multiwindow Gabor system is one of generalizations of multiwindow Gabor systems, whose time-frequency shifts vary with the windows
This paper addresses subspace mixed multiwindow Gabor systems with rational time-frequency product lattices

Summary

Introduction

Let H be a separable Hilbert space. An at most countable sequence {hi}i∈I in H is called a frame for H if there exist constants 0 < A ≤ B < ∞ such that. Gl(t, v)r,k = Zaqgl t – ra + , v b for r ∈ Nq, k ∈ Np. Remark 2.1 By the quasi-periodicity of Zaq, for an arbitrary f ∈ L2(R), Zaqf is uniquely determined by the values of (Zaqf )(·, ·) on S × [0, 1) with S being a set aqZ-congruent to [0, aq). HL} ⊂ L2(R) with G(h, a, b) being a Bessel sequence in L2(R), G(h, a, b) is a Gabor dual of type I (type II) for G(g, a, b) if and only if the following hold:. HL} ⊂ L2(R) with G(h, a, b) being a Bessel sequence in L2(R), we have (i) G(h, a, b) is a Gabor dual of type I for G(g, a, b) if and only if there exists a measurable.

Some examples and remarks

Conclusions