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Main Title: Measures of Scalability
Author(s): Chen, Xuemei
Kutyniok, Gitta
Okoudjou, Kasso
Philipp, Friedrich
Wang, Rongrong
Type: Research Paper
Abstract: Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.
Subject(s): convex geometry
quality measures
parseval frame
scalable frame
Issue Date: 10-Oct-2014
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2014, 24
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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