Unidimensional and bidimensional prolate spheroidal wave functions and applications

In this work, we give a brief description of the theory and the properties of the 1-D and 2-D prolate spheroidal wave functions (PSWFs). Then, we study some efficient methods for the computation of the different PSWFs and their associated eigenvalues. More precisely, we shall develop a quadrature method for the approximation of the 1-D PSWFs. Then, we extend this method to the construction of the 2-D PSWFs that are concentrated on the unit disk. Also, we give a brief review of the quality of approximation of almost band-limited functions by the 1-D PSWFs. Moreover, we use some numerical tests to compare between PSWFs and wavelets as signal processing tools. In particular, we give some numerical tests that highlight the efficiency of the PSWFs in processing band-limited signals.