Duality and Biorthogonality for Weyl-Heisenberg Frames期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Duality and Biorthogonality for Weyl-Heisenberg Frames

Authors:	AJEM Janssen

Institution:	(1) Philips Research Laboratories Eindhoven, 5656 AA Eindhoven, The Netherlands

Abstract:	Let and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$ $t\in{\Bbb R}$ , for any $f\in L^2({\Bbb R}).$ We show that is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$ $B<\infty.$ Next, when generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.

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