Mathematical Sciences: Orthogonal Polynomials

该项目计划的研究集中在 称为正交多项式的特殊函数。 正交性 只能在存在措施的情况下考虑;在这项工作中 这些措施将集中在真实的线上， 圈 这项工作的重点是措施之间的联系， 相应正交矩阵的递推系数 多项式 还将开展调查工作， 正交多项式展开式的可和性和收敛性 该方法涉及最小化多项式积分（ Christoffel函数），差分的渐近行为 方程和无穷三对角矩阵的特征值 这些相同的技术可以用于研究的解决方案， 中心力场中的薛定谔方程和 Christoffel函数与统计预测理论有关。 正交多项式形成特殊函数的子类 经过几个世纪的发展， 物理世界的基本方程。 他们是 通常表达简单，相对容易计算，并且通常 作为某些微分方程的基本解出现。

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.