Generalized Gramians: Creating frame vectors in maximal subspaces

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.