关于雅可比-指数权的正交多项式

Orthogonal polynomial is an important branch of approximation theory of functions. It plays an important role in developing numerical and analytical methods in modern mathematics, physics, and engineering. Recently, we put forward a new class of special weights, the Jacobi-exponential weights (the combination of the two best important weights: generalized Jacobi weights and the exponential weights). The aim of the project is to study, totally or partial solve the following questions for Jacobi-exponential weights:.(1) the distribution of zeros of orthogonal polynomials for Jacobi-exponential weights;.(2) the bounds on orthogonal polynomials for Jacobi-exponential weights and its asymptotics;.(3) the extension of some properties of orthogonal polynomials for Jacobi-exponential weights on (-1,1) to the unit ball;.(4) the application of orthogonal polynomials for Jacobi-exponential weights on numerical integration..To approach the above three problems, we will focus on studying the theory of orthogonal polynomials for Jacobi-exponential weights using interpolation approximation, algebraic theory, measure theory and wavelet analysis. In addition, the convergence and divergence of extremal polynomials, interpolation polynomials and orthogonal polynomials will be combined organically. This branch new approach is not only a breakthrough in theory, but also in practical application.

正交多项式是函数逼近论的一个重要分支，在现代数学、物理和工程中的数值分析方法的发展中发挥着极为重要的作用。近期，我们提出一类全新的权函数，雅可比-指数权（两类重要权的结合：广义雅可比权与指数权），本项目旨在研究、解决或部分解决下列几个问题：.（1）雅可比-指数权的正交多项式的零点分布状态；.（2）雅可比-指数权的正交多项式的界及其渐近性；.（3）将区间 (-1,1) 上得到的雅可比-指数权正交多项式的部分性质推广于单位球上；.（4）雅可比-指数权的正交多项式在数值积分方面的应用。.本项目将以关于雅可比-指数权的正交多项式性质的研究为主线，以插值理论、代数理论、测度论与小波分析的思想、方法和技术技巧为工具，将极值多项式、插值多项式与正交多项式的敛散性有机地融合于正交多项式性质的研究，达到推进上述三个问题有效解决的目的。这不仅在理论上和应用上对于上述分支有一定的突破，且在研究方法上是一种突破。