On Orthogonal Polynomials Of Several Variables And Application

Orthogonal polynomials of several variables play a very important role in many theoretic fields and applied fields. This dissertation is to discuss mainly the common zeros and the reproducing kernels of multivariate orthogonal polynomials, and the orthogonal polynomials on the unit sphere Sd-1.(1) We consider the common zeros of multivariate orthogonal polynomials under the Gauss-type case. Firstly, we obtain a Chebyshev's maximum principle for the maximal common zeros, which is the extension of the result of the univariate case to that of the multivariate case. Secondly, we give out an upper bound for the minimal zeros of the univariate orthogonal polynomials. It can also be extended to the multivariate case under the Gauss-type condition. Finally, we obtain an asymptotic property of the common zeros of the multivariate orthogonal polynomials.(2) We consider the n-th reproducing kernel of the multivarite orthogonal polynomials. In the first section, we give out a generalised minimum property of the n-th reproducing kernel of the multivariate orthogonal polynomials. It is an extension of (1.1), and an extension of the result in [12] under the multivariate case. In the second section, we consider an extremum problem, and attain its upper and lower bounds represented by the n-th reproducing kernel of the univariate orthogonal polynomials. It can also be extended to the multivariate case. The third section is about an recursion property of the n-th reproducing kernel. In the last sections, we attain the expressions for the n-th reproducing kernels of the polynomials orthogonal with respect to rotation invariant measures, Szego polynomials, and the biorthogonal rational functions.(3) We consider the orthogonal polynomials on the unit sphere Sd-1. Firstly, we construct a basis of the orthogonal homogeneous polynomials of degree n, withrespect to the weight function. Secondly, we construct an orthogonal trigonometric system and an quadrature formula on S1. Lastly, we construct a properly posed set of nodes for Lagrange interpolation on S2. It can be applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.(4) We consider a vector-valued quadratic extremal problem. It is shown thatit is connected with a sequence of bi-variate orthogonal polynomials. It can be regarded as the extension of those results in [56] under the higher dimension case.(5) We consider the little q-Jacobi-Sobolev polynomials {Qn(x)}n and the little g-Jacobi polynomials. We obtain the relative asymptotics(6) Following the method in [28], we discuss a kind of multivariate wavelets.Some simple properties are obtained. Finally, appendix 1 and appendix 2 are two extensions of the bivariate Bernstein basis functions of tensor-product form and trigonometric form respectively. Appendix 1 is based on the generalised tensor-product Poisson functions, and appendix 2 is based on a convergent positive linear operator sequence.