Mini-Workshop: Modern Applications of s-numbers and Operator Ideals

The main aim of this mini-workshop was to present and discuss some modern applications of the functional-analytic concepts of $s$-numbers and operator ideals in areas like Numerical Analysis, Theory of Function Spaces, Signal Processing, Approximation Theory, Probability on Banach Spaces, and Statistical Learning Theory

Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials

AbstractRecurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function w such that w′w is a rational function) are shown to be solutions of nonlinear differential equations with respect to a well-chosen parameter, according to principles established by D. Chudnovsky and G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in an + 1Pn + 1 (x) = xpn(x) − anpn − 1 (x) of the orthogonal polynomials related to the weight exp (− x44 − tx2) on R satisfy 4an3än = (3an4 + 2tan2 − n)(an4 + 2tan2 + n), and an2 satisfies a Painlevé PIV equation

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Heat kernel generated frames in the setting of Dirichlet spaces

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincar\'e inequality which lead to heat kernels with small time Gaussian bounds and H\"older continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting

Basic theory of one-parameter semigroups

Continuous one-parameter semigroups of bounded operators occur in many branches of mathematics, both pure and applied. The calculus of functions of one real variable can be formulated in terms of the translation semigroup, solutions of the equations connected with classical phenomena such as heat propagation are described by semigroups, and one-parameter groups and semigroups also describe the dynamics of quantum mechanical systems. Although semigroups occur in many other areas the development and scope of the general theory covered in this chapter is well illustrated by the foregoing examples. Hence we begin with a brief discussion of each of them