Abstract
A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.
Highlights
Recall that according to the Plancherel theorem the Fourier transform φ (s) = ∫ eitsφ (t) dt (1)R of a function φ ∈ L2(R) satisfies the Parseval identity ∫φ (s)2 ds = (t)2 dt (2)and the inverse Fourier transform is φ (t) e−itsφ (s) ds. (3)As is known a number of classical problems are reduced to searching for the distribution functions σ(⋅) (“spectral functions”) such that tohfefSutnieclttijoens singteλg(sr)al sIa(tλis)fi=es∫Rcegrtλa(isn)dpσr(esa)swsiigtnheadgcivoenndfiatimonilsy
We show that in the case I = R there exists a unique spectral function σ(s) = (1/2π)s, in which case the above Parseval identity turns into the classical one
We introduce the concept of the matrix-valued spectral function σ(s) for the vector-valued Fourier transform of a vector-function φ(t) = {φ1(t), φ2(t), . . . , φn(t)} ∈ L2(I, Cn), such that support of φj lies in Ij
Summary
At the same time we show that in the case I = R the set SF(R) consists of the unique spectral function σ(s) = (1/2π)s, for which equality (5) turns into the inverse Fourier–Plancherel transform (3) of a function φ ∈ L2(R). A matrix-valued distribution function σ : R → Cn×n will be called a spectral function for the vector-valued Fourier transform (1) (with respect to Ĩ) if the following Parseval identity holds:. S σF (s) = ∫ ΣF (x) dx give a bijective correspondence σ(⋅) = σF(⋅) between all functions F ∈ SR and all spectral functions σ(⋅) ∈ SF2((0, ∞), (−∞, 0)) These theorems show that the sets of spectral functions differ significantly for systems Ĩ formed by finite or infinite intervals. In conclusion note that the results of the paper on spectral functions for the classical Fourier transform are obtained with the aid of the results from [2,3,4] concerning spectral functions for canonical differential systems
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