ABSTRACT
This chapter discusses a linear vector space, a metric space, and a space of locally integrable functions. It also discusses imbedding theorems, and illustrates Sobolev spaces via the Fourier transform. The chapter proposes a distribution theory, and considers spaces of test functions and distributions. Every distribution supported at a point is a finite linear combination and its derivatives. The Paley-Wiener-Schwartz theorems are reviewed along with pseudodifferential and Fourier integral operators. Gegenbauer or ultraspherical polynomials, Legendre polynomials, Hermite polynomials, and Chebyshev polynomials are discussed. Laplace’s method is described for presenting the asymptotics of the integrals. The chapter provides a set of examples of regularization of ill-posed problems, and discusses random perturbation of the data. The Monte Carlo method (or method of statistical trials) is referred for solving various problems of computational mathematics by means of the construction of some random process for each problem, with the parameters of the process equal to the required quantities of the problem.
