Mathematical Sciences: Projects in Classical Analysis

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The central theme of this mathematical science research project group is approximation theory. Problems will be selected from areas such as differential equations, interpolation of operators, orthogonal polynomials and special functions. General areas of research include potential theoretic methods in approximation theory which has already provided the necessary tools for the resolution of two major conjectures in approximation theory. A second project concerns multivariate special functions including the evaluation of multivariate beta and q-beta type integrals and finding constant terms in Macdonald type identities. Asymptotic expansions, integral representations and generating functions of multivariate analogs of the classical orthogonal polynomials will be studied. Other work will focus on the behavior of best and near best approximants on compact plane sets. Specifically, efforts will be made to understand the sets where nearest point polynomials actually achieve the minimal distance (in the uniform norm.) Orthogonal polynomials play a key role in the spectral theory of second order difference and differential equations. This research will investigate the general fourth order equations. Explicit solutions and the computation of the spectral measure will be primary goals. A related line of investigation will consider the nature of roots of polynomials orthogonal with respect to absolutely continuous measures in cases where Markov's theorem does not apply.