Gaussian structures and orthogonal polynomials

University dissertation from Uppsala : Avdelningen för matematik

Abstract: This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like (p-1)-1 or like p when p approaches 1 or ? respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the Lp-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the Lp-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.