  multiplication operator on a Hilbert space may be approximated with finite
sections by choosing an orthonormal basis of the Hilbert space. Nonzero
multiplication operators on $L^2$ 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 well studied Toeplitz case. The use of circulant
approximations leads to constructive algorithms working for the separable
multivariate and matrix-valued cases as well.
