Vol. 2, No. 4, 2009

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Yet another generalization of frames and Riesz bases

Reza Joveini and Massoud Amini

Vol. 2 (2009), No. 4, 397–409

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.

$g$-frames, $g$-Riesz bases
Mathematical Subject Classification 2000
Primary: 42C15, 42C99
Secondary: 42C40
Received: 29 September 2008
Accepted: 19 March 2009
Published: 28 October 2009

Communicated by David Larson
Reza Joveini
Faculty of Science
Islamic Azad University of Bojnourd
Bojnourd 94176-94686
Massoud Amini
Faculty of Mathematical Sciences
Tarbiat Modares University
Tehran 14115-175
Institut Penyelidikan Matematik
Universiti Putra Malaysia
43400 UPM Serdang
Selangor Darul Ehsan, Malaysia