Spectral Functions for the Vector-Valued Fourier Transform

A scalar distribution function sigma(s) is called a spectral function for the Fourier transform phi(s) = integral(R) e(its) phi(t)dt (with respect to an interval F subset of R) if for each function phi is an element of L-2(R) with support in F the Parseval identity integral(R) |phi(s)|(2) do(s) = integral(R) |phi(t)|(2)dt holds. We show that in the case F = R there exists a unique spectral function sigma(s) (1 /2 pi)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval F = (0, b), there exist infinitely many spectral functions (with respect to F). We introduce also the concept of the matrix-valued spectral function sigma(s) (with respect to a system of intervals {F-1, F-2, ... F-n}) for the vector-valued Fourier transform of a vector-function phi(t) {phi(1)(t), phi(2) (t), ..., phi(n)(t) is an element of L-2(F, C-n), such that support of phi(j) lies in F-j. The main result is a parametrization of all matrix (in particular scalar) spectral functions sigma(s) for various systems of intervals {F-1, F-2, ..., F-n}.