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Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L 2
Author(s) -
Katsevich Alexander,
Bertola Marco,
Tovbis Alexander
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800567
Subject(s) - mathematics , hilbert matrix , uniqueness , inversion (geology) , hilbert transform , injective function , matrix (chemical analysis) , matrix function , pure mathematics , disjoint sets , mathematical analysis , symmetric matrix , hilbert space , eigenvalues and eigenvectors , quantum mechanics , paleontology , statistics , spectral density , materials science , physics , reproducing kernel hilbert space , structural basin , projective hilbert space , composite material , biology
Given n disjoint intervals Ijon R together with n functionsψj ∈ L 2 ( Ij ) , j = 1 , ⋯ n , and an n × n matrix Θ = ( θj k ) , the problem is to find an L 2 solutionφ ⃗ = Col ( φ 1 , ⋯ , φ n ) ,φj ∈ L 2 ( Ij ) , to the linear system χ Θ H φ ⃗ = ψ ⃗ , whereψ ⃗ = Col ( ψ 1 , ⋯ , ψ n ) , H = diag ( H 1 , ⋯ , H n ) is a matrix of finite Hilbert transforms with Hjdefined onL 2 ( Ij ) , and χ = diag ( χ 1 , ⋯ , χ n ) is a matrix of the corresponding characteristic functions on Ij . Since we can interpret χ Θ H φ ⃗ , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of R and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.