Super Oblique Gabor Duals of Super Gabor Frames on Discrete Periodic Sets

Abstract

The notion of superframe in general Hilbert spaces was introduced in the context of multiplexing, which has been widely used in mobile communication network, satellite communication network and computer area network. The notion of oblique dual frame is a generalization of conventional dual frame. It has provided us with a frame-like expansion. Using oblique dual frames one can extend frame expansions to include redundant expansions in which the analysis and synthesis frames lie in different spaces. Given positive integers L, M and N, an Nℤ-periodic set 𝕊 in ℤ, let 𝒢(g, N, M) be a frame for l ²(𝕊, ℂ ^L ), and let 𝒢(h, N, M) be a frame for ℒ(h, N, M) (generated by 𝒢(h, N, M)). This article addresses super Gabor duals of g in ℒ(h, N, M). We obtain a necessary and sufficient condition on h admitting super oblique Gabor duals of g, and present a parametrization expression of all super oblique Gabor duals and all oblique canonical Gabor duals of g. We also characterize the uniqueness of super oblique Gabor dual and oblique canonical Gabor dual of g. Some examples are also provided.

Keywords:

• Discrete periodic sets
• Discrete Zak transform
• Oblique Gabor dual
• Superframe
• Super Gabor frame

Mathematics Subject Classification:

• 42C15
• 42C40

ACKNOWLEDGMENTS

The authors would like to thank the referees for their helpful suggestions.

Supported by the National Natural Science Foundation of China (Grant No. 10901013, 11271037), Beijing Natural Science Foundation (Grant No. 1122008), the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030), the Fundamental Research Funds for the Central Universities (Grant No. 2011JBM299).