Functions With Disconnected Spectrum: Sampling, Interpolation, Translates.
Functions With Disconnected Spectrum: Sampling, Interpolation, Translates
Alexander M. Olevskii and Alexander Ulanovskii
American Mathematical Society
University Lecture Series; Volume 65
This volume contains 12 lectures on functions with disconnected spectrum, dealing with problems of Fourier analysis, which are partially inspired by signal theory. It discusses the classical sampling problem, the problem of interpolation of discrete functions by analytic ones with spectrum in S, and the problem of completeness of discrete translates. It provides an elementary introduction; classical results due to Beurling, Kahane, and Landau; recent progress in the area, such as construction of universal sampling sets and high-dimensional and non-analytic phenomena; and the completeness problem for discrete translates. It illustrates how methods of harmonic and complex analysis interact with various key concepts in different areas, such as Minkowski's lattice, Kolmogorov's width, Meter's quasicrystals, and others. ([umlaut] Ringgold, Inc., Portland, OR)