Potentials, Polynomials and Weighted Polynomial Inequalities

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The principal investigator explores several problems in approximation theory with special interest in applications of potentials and in polynomials. One of these topics is polynomial approximation with varying weights. Recently several problems have lead to this type of approximation, and the PI continues its study with the help of logarithmic potentials in the presence of an external field. This approximation problem is fundamental in proving different types of asymptotics for orthogonal polynomials and Christoffel functions with respect to varying weights. These in turn are related to some mechanical models of quantum physics, in particular to the universality hypothesis (saying that the distribution of energy levels in small intervals is independent of the original (random) distribution of the particles). Another related problem is on orthogonal polynomials with respect to a measure with compact support on the complex plane (bounds for polynomials, regularity criteria, Szego's problem on curves or disjoint intervals). Finally, the PI plans to investigate polynomial inequalities with as general weights as possible. Recently it has turned out that classical polynomial inequalities are also true in weighted form, and for a surprisingly wide class of weights, namely for so called doubling weights, or for weights satisfying the weakest of the Muckenhoupt A conditions. The PI further investigates weighted polynomial inequalities, as well as their consequences in other areas. This research project deals with some theoretical questions of mathematical analysis with the aim of using in more applied areas of mathematics and theoretical physics. Some basic problems in approximation theory are investigated with tools that have been traditionally associated with classical physics (potential theory). This novel application of potential theory makes it possible to attack problems that resisted classically applied tools, e.g. it makes possible to find the behav ior of some nonclassical orthogonal polynomials (Freud polynomials). A new impulse to the research came in recent years when it has been found that these orthogonal polynomials are related to some questions of theoretical physics that are modelling the uncertain nature of quantum systems by random mechanical models. Another major area of research is on weighted inequalities on polynomials. Polynomials are basic tools in replacing complicated structures by simpler ones that can be mathematically handled; and in this approximation process one needs various inequalities that relate various quantities of polynomials to one another.