Mathematical Sciences: Orthogonal Polynomials

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Work on this project will focus on topics in mathematical analysis connected with orthogonal polynomials. There are two directions the work takes. First one can begin with a measure on the real line or unit circle and consider the corresponding orthogonal polynomials and their recurrence relations. Conversely, one may begin with a class of polynomials arising from a recurrence relation and ask for information about measures against which they become orthogonal. In this latter context, the existence of measures yielding the recurrence coefficients has been known for some time. Work to be done here is that of determining the measure from the coefficients in a computationally efficient way. Another line of investigation will pursue the asymptotic behavior of the Christoffel function associated with a measure. Christoffel functions are related to statistical prediction theory. They minimize polynomial integrals associated with families of orthogonal polynomials. In addition, work will be done analyzing eigenvalues of infinite tri-diagonal matrices used in the study of Schrodinger equations in central force fields. Specific goals include improving an inequality for the limits of Christoffel functions on the unit circle associated with measures of the Szego class and narrowing the gap between the theory of orthogonal polynomials and signal processing. Results are expected to play an increasing role in the processing and transmission of electronic signals.