Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in Rn , but very little is known about the ℓ2( Z ) case or the ℓ2( Zd ) case. We establish some basic Gabor frame theory for ℓ 2( Z ) and then generalize to the ℓ2( Zd ) case.
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