替代正交多项式

Orthogonal polynomials on planar cubic curves

Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

Orthogonal polynomials in and on a quadratic surface of revolution

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

替代雅可比多项式与正交指数函数

Alternative Jacobi Polynomials and Orthogonal Exponentials

定义了区间[0,1]上替代雅可比多项式的正交多项式序列，并建立了其性质；提出了半轴[0,∞)上指数函数的(α,β)参数化正交多项式系统，并描述了两种子系统和正交指数多项式，同时引入了区间[0,1]上两个参数化的离散几乎正交函数系统。

Weighted Lagrange Interpolation Using Orthogonal Polynomials: Stenger's Conjecture, Numerical Approach

In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error estimation for such kinds of approximations. Later, we introduce the sufficient condition of Stenger's conjecture for orthogonal polynomials and numerical verification for such conjecture.

通过无限维带状矩阵分解实现正交多项式的多项式与有理测度修改

Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations

本文提出利用无限维带状矩阵分解的快速算法，以线性复杂度计算正交多项式在多项式或有理测度修改下的连接系数与修改Jacobi矩阵，并构造支持带状微分矩阵的正交多项式族以实现稀疏谱方法。

Fast associated classical orthogonal polynomial transforms

We discuss a fast approximate solution to the associated classical -- classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue problem with polynomial coefficients such that the differential operator is degree-preserving. Upon linearization, the discretization of this quadratic eigenvalue problem is block upper-triangular and banded. After a perfect shuffle, we extend a divide-and-conquer approach to the upper-triangular and banded generalized eigenvalue problem to the blocked case, w…