Sobolev orthogonality and differential equations

Abstract

During the last years, there has been a considerable amount of research on the subject of Sobolev orthogonal polynomials, which deals with the study of sequences of polynomials which are orthogonal relative to Sobolev inner product of the type ◁수식 삽입▷(원문을 참조하세요) where each $dμ_i$ is a signed Borel measure on the real line with finite moments. The reason which these polynomials has attracted the attention of many researchers is of their intimate connection with spectral theory of ordinary differential equations, Fourier expansions, smooth data fitting, behavior(or location) of their zeros as well as the comparison with the theory of standard orthogonal polynomials. In this thesis, we will first study the algebraic properties for these polynomials. This area deals with the study of an explicit representation, recurrence relations, and difference-differential relations for such polynomials. We then study differential equations with polynomials coefficients of three forms satisfied by such polynomials. In particular, we focus our attention on the spectral-type differential equations of order D ◁수식 삽입▷(원문을 참조하세요) satisfied by Sobolev orthogonal polynomials relative to π with N=1 in $(*)$,where $ℓ_i(x)$ is polynomial, independent of $n$ and $\lambda_n$ is an eigenvalue parameter. In this case, we find necessary and sufficient conditions for the differential equation $(**)$ to have such orthogonal polynomial solutions and also discuss the structure of distributional orthogonalizing weights for such polynomials and symmetrizability of such differential operators. This result not only generalizes a result by H. L. Krall, which handles the case when $dμ_1 \equiv 0$ but also provides a valuable technique in constructing weight functions for certain sequences of orthogonal polynomials.