Orthonormal Series Expansions of Certain Distributions and Distributional Transform Calculus

摘要

A technique for expanding certain Schwartz distributions into series of orthonormal functions is devised. The method works for all the classical orthogonal polynomials and many other sets of orthogonal functions. This result is then used to generalize various standard integral transforms, which are based on orthogonal series expansions, to distributions. As specific examples, the following distributional transforms are developed: the finite Fourier transform, the Laguerre transform, the Hermite transform, the Jacobi transform, the Legendre transform, the Chebyshev transform, the Gegenbauer transform, the finite Hankel transform of zero order. An application to the solution of differential equations is given. (Author)