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Title of Journal: Acta Appl Math

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Abbravation: Acta Applicandae Mathematicae

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Springer Netherlands

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DOI

10.1002/dat.20551

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1572-9036

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Duals of Frame Sequences

Authors: Christopher Heil Yoo Young Koo Jae Kun Lim
Publish Date: 2008/12/16
Volume: 107, Issue: 1-3, Pages: 75-90
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Abstract

Frames provide unconditional basislike but generally nonunique representations of vectors in a Hilbert space H The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations In particular oblique duals Type I duals and Type II duals have been introduced in the literature because of the special properties that they possess A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence This paper proves that all Type I and Type II duals are oblique duals but not conversely and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H as well as characterizing when the Type I Type II and oblique duals will be unique These results are also applied to the case of shiftgenerated sequences that are frames for shiftinvariant subspaces of L 2ℝ d

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