Generalized Gramians: Creating frame vectors in maximal subspaces

TL;DR

This paper introduces a generalized approach to frames in Hilbert spaces, removing traditional restrictions on Gramian matrices and enabling direct integral formulas, with applications to reproducing kernel spaces and random fields.

Contribution

It extends frame theory by removing restrictions on Gramian matrices and developing direct-integral analysis/synthesis formulas for spectral subspaces.

Findings

Existence of standard frames in spectral subspaces with bounds equal to interval endpoints

Application of generalized frames to reproducing kernel Hilbert spaces

Application to analysis of random fields

Abstract

A frame is a system of vectors $S$ in Hilbert space $\mathscr{H}$ with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to $S$, for all vectors in $\mathscr{H}$; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an $S$-Gramian, $G_{S}$ (an infinite matrix with entries equal to the inner product of pairs of vectors in $S$); but still with strong restrictions on the given system of vectors in $S$, in order to guarantee frame-bounds. In this paper we remove these restrictions on $G_{S}$, and we obtain instead direct-integral analysis/synthesis formulas. We show that, in spectral subspaces of every finite interval $J$ in the positive half-line, there are associated standard frames, with frame-bounds equal the endpoints of $J$. Applications are given to reproducing kernel Hilbert spaces, and to random fields.