A duality principle for groups II: Multi-frames meet super-frames

Abstract

The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor representations $π_Λ$ and $π_{Λ^{o}}$ with respect to a pair of dual time-frequency lattices $Λ$ and $Λ^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{π_Λ(m, n)g_{1} \oplus ... \oplus π_Λ(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{π_{Λ^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{π_Λ(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{π_{Λ^{o}}(m, n)g_{1} \oplus ... \oplus π_{Λ^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.