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Preprint 24-2014

Measures of Scalability

Source file is available as :   Portable Document Format (PDF)

Author(s) : Xuemei Chen , Gitta Kutyniok , Kasso Okoudjou , Friedrich Philipp , Rongrong Wang

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 24-2014

MSC 2000

42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions

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.

Keywords : Convex Geometry, Quality Measures, Parseval frame, Scalable frame

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