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2009 Yet another generalization of frames and Riesz bases
Reza Joveini, Massoud Amini
Involve 2(4): 397-409 (2009). DOI: 10.2140/involve.2009.2.397

Abstract

A frame is a sequence of vectors in a Hilbert space satisfying certain inequalities that make it valuable for signal processing and other purposes. There is a formula giving the reconstruction of a signal (a vector in the space) from its sequence of inner products (the Fourier coefficients) with the elements of the frame sequence. A g-frame, or operator-valued frame, is a sequence of operators defined on a countable ordered index set that has properties analogous to those of a frame sequence.

We present a new approach to the matter of defining a Hilbert space frame, indexed by an ordered set, when the set is a measure space which is not necessarily purely atomic. Continuous frames have been widely studied in the literature, but the measure spaces they are associated with are not necessarily ordered in any way. Our approach is to make the measure space a directed set, and then replace the sequence of vectors (or operators) with a net indexed by the directed set, obtaining a natural generalization of the usual notion of generalized frame. We show that this definition makes sense mathematically, and proceed to obtain generalizations of several of the standard results for frame and Bessel sequences, and also Riesz bases, g-frames and operator-valued frames.

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Reza Joveini. Massoud Amini. "Yet another generalization of frames and Riesz bases." Involve 2 (4) 397 - 409, 2009. https://doi.org/10.2140/involve.2009.2.397

Information

Received: 29 September 2008; Accepted: 19 March 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1184.42026
MathSciNet: MR2579559
Digital Object Identifier: 10.2140/involve.2009.2.397

Subjects:
Primary: 42C15, 42C99
Secondary: ‎42C40

Rights: Copyright © 2009 Mathematical Sciences Publishers

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Vol.2 • No. 4 • 2009
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