NDLI: Duals of Frame Sequences

Content Provider	Springer Nature Link
Author	Koo, Yoo Young Heil, Christopher Lim, Jae Kun
Copyright Year	2008
Abstract	Frames provide unconditional basis-like, 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 shift-generated sequences that are frames for shift-invariant subspaces of L 2(ℝ d ).
Ending Page	90
Page Count	16
Starting Page	75
File Format	PDF
ISSN	01678019
e-ISSN	15729036
Journal	Acta Applicandae Mathematica
Issue Number	1-3
Volume Number	107
Language	English
Publisher	Springer Netherlands
Publisher Date	2008-12-16
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Frame Oblique projection Mathematical and Computational Physics Type I dual Mathematics Perturbation Gramian operator Angle between subspaces Frame sequence Computer Science Mechanics General harmonic expansions, frames Type II dual Oblique dual Riesz basis Statistical Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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