Pseudospectral functions of various dimensions for symmetric systems with the maximal deficiency index

Mogilevskii, Vadim

doi:10.1007/s10958-018-3662-y

Cited by 1 publication

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“…(here is the orthoprojector in C onto C − ). A description of all spectral functions of system (20) is given by the following theorem, which directly follows from Theorem 4.17 and formulas (4.68)-(4.71) in [3].…”

Section: Canonical Systemsmentioning

confidence: 99%

“…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. We denote by S(C , C ) the set of all holomorphic operator-functions : C + → B(C , C ) such that ‖ ( )‖ ≤ 1, ∈ C + (the latter means that the function is contractive).…”

Section: Introductionmentioning

confidence: 99%

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Spectral Functions for the Vector-Valued Fourier Transform

Mogilevskii

2018

Journal of Function Spaces

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A scalar distribution function ( ) is called a spectral function for the Fourier transform̂( ) = ∫ R ( ) (with respect to anWe show that in the case I = R there exists a unique spectral function ( ) = (1/2 ) , in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I = (0, ), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function ( ) (with respect to a system of intervals {I 1 , I 2 , . . . , I }) for the vector-valued Fourier transform of a vector-function ( ) = { 1 ( ), 2 ( ), . . . , ( )} ∈ 2 (I, C ), such that support of lies in I . The main result is a parametrization of all matrix (in particular scalar) spectral functions ( ) for various systems of intervals {I 1 , I 2 , . . . , I }.

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Section: Canonical Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%