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

Given n disjoint intervals Ij on R together with n functions ψj∈L2(Ij), j=1,⋯n, and an n×n matrix Θ=(θjk), the problem is to find an L2 solution φ⃗=Col(φ1,⋯,φn), φj∈L2(Ij), to the linear system χΘHφ⃗=ψ⃗, where ψ⃗=Col(ψ1,⋯,ψn), H=diag(H1,⋯,Hn) is a matrix of finite Hilbert transforms with Hj defined on L2(Ij), and χ=diag(χ1,⋯,χn) is a matrix of the corresponding characteristic functions on Ij. Since we can interpret χΘHφ⃗, 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.