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Scalable Frames

Please always quote using this URN:urn:nbn:de:0296-matheon-11597
  • Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.

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Metadaten
Author:Gitta Kutyniok, Kasso A. Okoudjou, Friedrich Philipp, Elizabeth K. Tuley
URN:urn:nbn:de:0296-matheon-11597
Referee:Volker Mehrmann
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/07/27
Release Date:2012/07/27
Tag:Conical surfaces; Diagonal operator; Preconditioner; Tight frames
Institute:Technische Universität Berlin
MSC-Classification:12-XX FIELD THEORY AND POLYNOMIALS / 12Dxx Real and complex fields / 12D10 Polynomials: location of zeros (algebraic theorems) (For the analytic theory, see 26C10, 30C15)
14-XX ALGEBRAIC GEOMETRY / 14Pxx Real algebraic and real analytic geometry / 14P05 Real algebraic sets [See also 12Dxx]
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A03 Vector spaces, linear dependence, rank
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A12 Conditioning of matrices [See also 65F35]
42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C15 General harmonic expansions, frames
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F08 Preconditioners for iterative methods
Preprint Number:974
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