A Hilbert Space Approach to Some Classical Transforms: (196 Pitman Research Notes in Mathematics Series)

The following notes are based on a course held at the University of Bonn 1983/4 and are an extended version of a previous manuscript with the title "On Some Special Functions of Mathematical Physics and Their Relation to Spectral Theory". One idea on which the study is based is the observation that some of the well-known integral transforms are closely connected with the spectral theory of a differential operator. These integral transforms can in fact be "guessed" from the spectral resolution of an appropriately defined selfadjoint realization of this differential operator without using the machinery of Titchmarch-Weyl-Kodaira theory. In the light of this approach, the discussion of many integral transforms may be considered as immediate application of general spectral theory of selfadjoint operators. The main emphasis is however on reconstructing special integral transforms, such as the Laplace transform and the Hankel transform as spectral theorems. The integral transforms are considered as constructive versions of an abstract spectral representation, the existance of which is assured by general spectral theory of selfadjoint operators. These lecture notes are not intended as a replacement of investigations of special function of mathematical physics but rather an approach of some aspects of this area from a certain point of view. The emphasis of the notes is on ideas rather than techniques and the approach is to establish the means to bring order into the zoo of special functions. The application of abstract spectral theory to special examples are used as a source of examples and exercises in functional analysis.

Table of Contents:
Part 1 Spectral representation results related to the differential expression D: the Hilbert space approach for the differential operator D; selfadjoint realizations of the operator D in the case of a bounded interval - the Fourier series; basic properties of the Fourier transform; tempered Sobolev spaces; the Hartley transform; functions of D in the case of an unbounded interval; differential operators with constant coefficients; convolutions; Hilbert transform; Abel transform; the sampling theorem for the Fourier transform; on almost periodic functions; the higher dimensional Fourier transform as a spectral representation; the Gauss-Weierstrass transform; the Laplace transform; the Mellin transform. Part 2 Spectral results related to some formally selfadjoint differential expressions of second order: some polynomia eigensolutions - Legendre polynomials, Gegenbauer polynomials, Jacobi polynomials, Chebycheff polynomials, Hermite polynomials, Laguerre polynomials; the Lebedev transform; the Hankel transform; Mehler transfrom; the Jacobi transform. Appendix: tools from functional analysis.