Yet another generalization of frames and Riesz bases

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.