  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.
