Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L2

Abstract

AbstractGiven n disjoint intervals on together with n functions , , and an matrix , the problem is to find an L2 solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , 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 and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.