Continuous frames in Hilbert space. (English) Zbl 0782.47019

Ali, S. Twareque; Antoine, J.-P.; Gazeau, J.-P.

Ann. Phys. 222, No. 1, 1-37 (1993).

Summary: 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 relationship 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 affine group.

MSC:

47A70	(Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47N50	Applications of operator theory in the physical sciences
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

frames in Hilbert spaces; reproducing kernel Hilbert spaces