Continuous Frames in Hilbert Space

Abstract

The standard theory of frames in Hilbert spaces, using discrete bases, is generalized to one where the basis vectors may be labelled using discrete, continuous, or a mixture of the two types of indices. A comprehensive analysis of such frames is presented and various notions of equivalence among frames are introduced. A consideration of the relationschip between reproducing kernel Hilbert spaces and frames leads to an exhaustive construction for all possible frames in a separable Hilbert space. Generalizations of the theory are indicated and illustrated by an example drawn from the afline group.