Diagonalization of the finite Hilbert transform on two adjacent intervals : the Riemann-Hilbert approach

• In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms H-L : L-2([b(L), 0]) -> L-2([0, b(R)]) and H-R : L-2([0, b(R)]) -> L-2([b(L), 0]). These operators arise when one studies the interior problem of tomography. The diagonalization of H-R, H-L has been previously obtained, but only asymptotically when b(L) not equal -b(R). We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes H-R, H-L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

Ämnesord

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

Diagonalization of integral operators with continuous spectrum

Finite Hilbert transforms on multi intervals

Spectral theory of finite Hilbert transforms

Riemann-Hilbert problem

Asymptotics of hypergeometric functions

Publikations- och innehållstyp

ref (ämneskategori)

art (ämneskategori)