Mathematical Sciences: Orthogonal Polynomials

这个项目计划的研究集中在一类被称为正交多项式的特殊函数上。只有在存在测量的情况下才能考虑正交性；在这项工作中，测量将集中在真正的直线上或圆周上。研究了一种度量与相应的正交多项式的递归系数之间的关系。还将研究正交多项式展开式的可求和性和收敛问题。这些方法涉及最小化多项式积分(Christoffel函数)、差分方程组的渐近行为和无穷三对角阵的特征值。这些技术同样可以用于研究中心力场中薛定谔方程的解，而克里斯托弗函数与统计预测理论有关。正交多项式形成了几个世纪以来发展起来的特殊函数的子类，用于求解或近似求解模拟物理世界的基本方程。它们通常表达简单，计算相对容易，并且通常作为某些微分方程式的基本解出现。

The research planned in this project centers on classes of special functions known as orthogonal polynomials. Orthogonality can only be considered in the presence of a measure; in this work the measures will be concentrated on the real line or on a circle. The work focuses on the connection between a measure and the recurrence coefficients of the corresponding orthogonal polynomials. Work will also be done investigating the summability and convergence of orthogonal polynomial expansions. The methods involve minimizing polynomial integrals (the Christoffel function), asymptotic behavior of difference equations and eigenvalues of infinite tridiagonal matrices. These same techniques can be used for studying solutions of the Schrodinger equation in a central field of force and the Christoffel function is related to statistical prediction theory. Orthogonal polynomials form a subclass of special functions developed over centuries for the solution or approximate solution of fundamental equations modeling the physical world. They are generally simple to express, relatively easy to compute and often occur as fundamental solutions of certain differential equations.