Generalized Gramians: Creating frame vectors in maximal subspaces

Abstract

A frame is a system of vectors [Formula: see text] in Hilbert space [Formula: see text] with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to [Formula: see text], for all vectors in [Formula: see text]; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an [Formula: see text]-Gramian, [Formula: see text] (an infinite matrix with entries equal to the inner product of pairs of vectors in [Formula: see text]); but still with strong restrictions on the given system of vectors in [Formula: see text], in order to guarantee frame-bounds. In this paper, we remove these restrictions on [Formula: see text], and we obtain instead direct-integral analysis/synthesis formulas. Applications are given to reproducing kernel Hilbert spaces, and to random fields.