Abstract
The main purpose of the paper is to give a characterization of all compactly supported dual windows of a Gabor frame. As an application, we consider an iterative procedure for approximation of the canonical dual window via compactly supported dual windows on every step. In particular, the procedure allows to have approximation of the canonical dual window via dual windows from certain modulation spaces or from the Schwartz space.
Highlights
Introduction and main resultsThroughout the paper, H denotes a separable Hilbert space
{SG−1 gk }∞ k=1 is a dual frame of the frame G, called the canonical dual of G
Given g ∈ L2(R) and positive constants a and b, a Gabor system is a system of the form {EmbTnag}m,n∈Z; g is called the window of the system {EmbTnag}m,n∈Z
Summary
Throughout the paper, H denotes a separable Hilbert space. A sequence G = {gk}∞ k=1 with elements from H is a frame for H [15], if there exist positive constants AG and BG such that. The main purpose of this paper is to give a characterization of all the dual windows which have compact support and to avoid operator inversions: Theorem 1.1 Let g ∈ L2(R) be compactly supported and such that G = {EmbTnag}m,n∈Z is a frame for L2(R) for some a, b > 0. As an application of Theorem 1.1, we can give a procedure for approximation of the canonical dual window via compactly supported dual windows (Proposition 1.2). We consider an algorithm (variation of the frame algorithm), which provides compactly supported dual windows on every iteration and applies to a quite general class of Gabor frames.
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