【作者】 周恒；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2004， 博士

【摘要】 开展多元正交多项式的研究既有深刻的理论价值，也有着广泛的应用前景。本文主要是针对多元正交多项式的公共零点和再生核，以及单位球面S^d-1上正交多项式的一些研究工作。主要工作如下： （1）考虑Gauss-型线性泛函下多元正交多项式的公共零点。首先，得到了多元正交多项式最大公共零点的一个Chebyshev最大原理。它是一元正交多项式最大零点Chebyshev最大原理的一个推广。齐次，我们从一元的情形出发，给出了一元正交多项式最小零点的一个上界估计。它可以推广到Gauss-型线性泛函的多元情形。最后，我们得到了关于多元正交多项式递推关系系数矩阵的公共零点的一个渐近性质。 （2）考虑多元正交多项式的再生核。第一节我们给出了多元正交多项式再生核的一个广义极小性质。它是（1．1）的一种推广，也可看作[12]中的结果在多元情形下一个推广。第二节主要讨论了一元情形下的一个极值问题的估值，得到了由再生核表示的上下界估计。它也有多元情形的相应结果。第三节则是多元正交多项式再生核一个“递归”性质的讨论。而在第四，五，六节中，我们分别得到了旋转不变权函数情形下二元正交多项式，Szeg多项式，及双正交有理函数第n次再生核的表达式。其中关于Szeg多项式和双正交有理函数的结果虽然不是新的，但我们这里采用的证明方法更加简单和明了。 （3）关于单位球面S^d-1上的正交多项式的讨论。构造了S^d-1上关于权函数的齐次正交多项式的一个基底；给出了S¹上的一个三角正交多项式系和一个求积公式；得到了S²上的一个Lagrange插值适定结点组，它可以用来构造п_n^d上的Lagrange插值适定结点组。 （4）讨论一个向量值二次极值问题与二元正交多项式的关系。得到，使得该极值问题取得一个极值的每个点必对应着一个二元正交多项式系在该极值处的值。它是[56]中结果在高维时的推广。 （5）关于一元Sobolev正交多项式的一个结果。考虑了q-差分内积中的小q-Jacobi-Sobolev多项式{Q_n（x）}_n和小q-Jacobi多项式{P_n^{（α-1，β-1）}（x）}n，得到了和的相对渐近性质。 （6）遵循[28]中的方法，利用多元正交多项式的某些性质，讨论了一种多元小波的一些简单性质.最后，附1和附2给出了二元张量积形式和三角形式Bernstein基函数的两个推广.附1中的推广是基于广义的张量积Poisson函数，而附2中的推广则是基于一个收敛的正线性算子序列.更多还原

【Abstract】 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.更多还原