Duality and Biorthogonality for Weyl-Heisenberg Frames

Abstract

Let $a>0, b>0, ab<1;$ 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 $H$ 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 $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ 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 $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$ $B<\infty.$ Next, when $g$ 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 $g$ 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.