ИСТИНА

The talk will review classical results about truncations of Toeplitz operators --- Szego's theorem and extensions, and band-limited Fourier transform (Landau-Pollack-Slepian theorem). Connections with limit laws of probability theory and with author's work on the symbol of the Dirichlet-to-Neumann operator in diffraction theory will be discussed. A typical problem of interest is of singular perturbation type. The unperturbed (or limiting) operator has continuous spectrum or simply is unitary. The points of its spectrum are approximated by a part of the eigenvalues of a "perturbed" (i.e. approximating) operator, which is compact or even finite-dimensional. At the same time, the approximating operator has many eigenvalues that correspond to "non-essential" degrees of freedom and quickly approach zero. Interesting asymptotic laws describe the eigenvalue distribution of the approximating operator in the transition zone between the two series of eigenvalues.