A Duality Principle for Groups II: Multi-frames Meet Super-Frames

Abstract

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) 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 biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that {pi _{Lambda }(m, n)g_{1} oplus cdots oplus pi _{Lambda }(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d})oplus cdots oplus L^{2}({mathbb {R}},^{d}) if and only if cup _{i=1}^{k}{pi _{Lambda ^{o}}(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a Riesz sequence, and cup _{i=1}^{k} {pi _{Lambda }(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d}) if and only if {pi _{Lambda ^{o}}(m, n)g_{1} oplus cdots oplus pi _{Lambda ^{o}}(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a Riesz sequence, where pi _{Lambda } and pi _{Lambda ^{o}} is a pair of Gabor representations restricted to a time–frequency lattice Lambda and its adjoint lattice Lambda ^{o} in {mathbb {R}},^{d}times {mathbb {R}},^{d}.