Spectral Functions for the Vector-Valued Fourier Transform. (1st October 2018) - Mogilevskii, Vadim
Abstract : A scalar distribution functionσ ( s ) is called a spectral function for the Fourier transformφ ^ ( s ) = ∫ R e i t s φ ( t ) d t (with respect to an intervalI ⊂ R ) if for each functionφ ∈ L 2 ( R ) with support inI the Parseval identity∫ R φ ^ s 2 d σ ( s ) = ∫ R φ t 2 d t holds. We show that in the caseI = R there exists a unique spectral functionσ ( s ) = ( 1 / 2 π ) s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite intervalI = ( 0, b ), there exist infinitely many spectral functions (with respect toI ). We introduce also the concept of the matrix-valued spectral functionσ ( s ) (with respect to a system of intervals{ I 1, I 2, …, I n } ) for the vector-valued Fourier transform of a vector-functionφ ( t ) = { φ 1 ( t ), φ 2 ( t ), …, φ n ( t ) } ∈ L 2 ( I, C n ), such that support ofφ j lies inI j . The main result is a parametrization of all matrix (in particular scalar) spectral functionsσ ( s ) for various systems of intervals{ I 1, I 2, …, I n } .