Probabilistic methods in problems of applied analysis

The use of probability methods in analysis is not new. One of the most elegant examples is Bernsteins's proof of Weierstrass' theorem (Section 3); one of the more ill-famed the suspiciously accurate approximation of $\pi$ by Lazzerini using Buffon's needle (cf. Kendall and Moran (1962)). For more recent applications we refer to Byrnes e.a. (1992). In this paper we present some, mostly rather recent, situations where methods from probability theory have been used to prove results in non-stochastic problems of (applied) analysis. Sometimes a problem in analysis can be put in a probabilistic context and then solved by standard probability methods, sometimes the solution depends on analytic properties that are well-known in probability theory but not in analysis. Here I shall give applications from three areas of probability that I have been involved with myself; brief applications of the law of large numbers and of infinite divisibility, and a more extensive application involving Poisson processes and the central limit theorem. The three areas of application are approximation theory, deconvolution of positive functions, and a description and solution of a problem concerning of the breakdown of insulator gases by ionization. In Section 2 we briefly review some probability concepts, in Sections 3,4 en 5 we discuss the three applications listed above