The spectral approximation of multiplication operators via asymptotic (structured) linear algebra

A multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Multiplication operators with nonzero symbols, defined on L² spaces of functions, are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol (i.e. the multiplier). We discuss the case of real orthogonal polynomial bases and the relations with the classical Fourier basis whose choice leads to the well studied Toeplitz case. Indeed, the asymptotic approximation of the spectrum by the spectra of the associated Toeplitz sections is possible only under precise geometric assumptions on the range of the symbol. Conversely, the use of circulant approximations leads to constructive algorithms, with O(N log(N)) complexity (N = number of sections), working in general and generalizable to the separable multivariate and matrix-valued cases as well.