Orthogonal Polynomials For Jacobi-exponential Weights On [-1,1]

Let I=(a,b) and W=e-Q, where Q:I→[0,∞] is continuous. Let U(x)=Πi=1r|x-ti|pi,0<p<∞,-∞≤a≤tr<tr-1<…<t2<t1≤b≤∞,r≥2,pi>-1/p, i=1,2,..., r. Recently, Shi [70] defined weights:Jacobi-exponential weights UW (the combination of the two best important weights:Jacobi weight U and the exponential weight W). In this paper, several properties of orthogonal polynomials for this new special weights are obtained. Let pn(x)=pn((UW)2,x) be the nth orthonormal polynomial with respect to the Jacobi-exponential weights. This article consists of five chapters and Appendices1and2.In Chapter1, the background and current situation of the orthogonal polyno-mials with respect to general weights and exponential weights are introduced. At the same time, some basic definitions, notations and theorems are provided. Finally, the main results of this article are given.In Chapter2. the restricted range inequalities for Jacobi-exponential weights on [-1.1] are studied. For U(x). we first separate two cases:-1≠r,1≠t1and-1=tr,1=t1]. and then get Lp analogues of the Mhaskar-Saff inequality, respectively. The difficulty of the section is how to treat the endpoints±＋1.We distinguish two subcases to study:i) p1,pr>0;ⅱ)a general case without the assumption that p1, pr≥0. In the second case we introduce the weight Q*(x) Q(x)+min{p1,pr,0}-ln(1-x2), W*(x):=e-Q*(x).In addition, we also give the relationship between the classes of W and W*.In Chapter3, we first give various technical estimates and then consider the generalized Christoffel functions for Jacobi-exponential weights on [-1,1] for all cases as in Chapter2. Using the main ideas in [70] with modifications,we get the estimates of Lp Christoffel functions for UW and generalized Christoffel functions with respect to W*.In Chapter4, we describe the distribution of the zeros of orthogonal polynomials Pn((UW)2,x) in detail. First, using the results of λn(UW:x) we obtain an upper bound for the spacing between the zeros. Next, we estimate the largest/smallest zeros. We can see that in comparison with the zeros of orthogonal polynomials for exponential weights or for Jacobi weights, things become more complicated for Jacobi-exponential weights, but the distributions of the zeros are similar. Finally, an example satisfying the above important results is given.In Chapter5, we give a summary of this paper and point out the possibilities of the topic for further study.Finally, Appendices1and2discuss the convergence for Hcrmite interpolation and uniqueness theorem of a system of nonlinear equations as well as its applica-tions, respectively, since orthogonal polynomials are very closely related to Hermite interpolation, Birkhoff interpolation, Gaussian quadrature formula, and power or-thogonal polynomials...