Spectral Approximation of Multiplication Operators
Kent E. Morrison · 1995
. A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. For an operator that is not compact such approximations cannot converge in the norm topology on the space of operators. Multiplication operators on spaces of L 2 functions are never compact; for them we consider how well the eigenvalues of the matrices approximate the spectrum of the multiplication operator, which is the essential range of the multiplier. The choice of the orthonormal basis strongly affects the convergence. Toeplitz matrices arise when using the Fourier basis of exponentials exp(ik`). We also consider the basis of Legendre polynomials and the basis of Walsh functions. Contents 1. Introduction 75 2. Multiplication operators 78 2.1. Toeplitz Matrices 78 2.2. Matrices Associated to Legendre Polynomials 78 2.3. Walsh-Toeplitz Matrices 79 3. Spectral Convergence 81 4. Spectral convergence for Toeplitz matrices 83 5. Spectral Convergence wit...