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Frame multiplication theory for vect...

Andrews, Travis D.

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Frame multiplication theory for vector-valued harmonic analysis.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Frame multiplication theory for vector-valued harmonic analysis./
Author:	Andrews, Travis D.
Description:	115 p.
Notes:	Source: Dissertation Abstracts International, Volume: 75-11(E), Section: B.
Contained By:	Dissertation Abstracts International75-11B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3627613
ISBN:	9781321030167

Frame multiplication theory for vector-valued harmonic analysis.
Andrews, Travis D.

Frame multiplication theory for vector-valued harmonic analysis. - 115 p.

Source: Dissertation Abstracts International, Volume: 75-11(E), Section: B.

Thesis (Ph.D.)--University of Maryland, College Park, 2014.

This item must not be sold to any third party vendors.

A tight frame is a sequence in a separable Hilbert space satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for a frame is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames.

ISBN: 9781321030167Subjects--Topical Terms:

515831
Mathematics.

Frame multiplication theory for vector-valued harmonic analysis.
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020 $a 9781321030167
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100 1 $a Andrews, Travis D. $3 2070407
245 1 0 $a Frame multiplication theory for vector-valued harmonic analysis.
300 $a 115 p.
500 $a Source: Dissertation Abstracts International, Volume: 75-11(E), Section: B.
500 $a Adviser: John J. Benedetto.
502 $a Thesis (Ph.D.)--University of Maryland, College Park, 2014.
506 $a This item must not be sold to any third party vendors.
520 $a A tight frame is a sequence in a separable Hilbert space satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for a frame is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames.
520 $a The motivation for this work is the desire to define meaningful vector-valued versions of the discrete Fourier transform and the discrete ambiguity function. We make these definitions and prove several familiar harmonic analysis results in this context. These definitions beget the questions we answer through developing frame multiplication theory.
520 $a For certain binary operations, those with the Latin square property, we give a characterization of the frames, in terms of their Gramians, that have these frame multiplications. Combining finite dimensional representation theory and Naimark's theorem, we show frames possessing a group frame multiplication are the projections of an orthonormal basis onto the isotypic components of the regular representations. In particular, for a finite group G, we prove there are only finitely many inequivalent frames possessing the group operation of G as a frame multiplication, and we give an explicit formula for the dimensions in which these frames exist. Finally, we connect our theory to a recently studied class of frames; we prove that frames possessing a group frame multiplication are the central G-frames, a class of frames generated by groups of operators on a Hilbert space.
590 $a School code: 0117.
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a University of Maryland, College Park. $b Mathematics. $3 1266601
773 0 $t Dissertation Abstracts International $g 75-11B(E).
790 $a 0117
791 $a Ph.D.
792 $a 2014
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3627613