Mathematical Sciences: Orthogonal Polynomials

数学分析中的各种主题 正交多项式将在本项目中讨论。 上 在同一区间内，有许多类正交多项式。 在每个类中，正交性是相对于一个测度定义的 （密度）集中在间隔上。 这一主题 工作是一个持续的努力，以了解关系， 在测量和递归系数之间， 对应的多项式。 关联到每个类的 正交多项式是一种独特的算法，它产生 高阶多项式的形式。 此次复发 是一个简单的线性关系，它是 这是一个相当有趣的关系。 方法包括最小化多项式 积分（Christoffel函数）， 差分方程与无穷三对角矩阵的特征值 矩阵 矩阵特征值的研究方法 可以适用于研究薛定谔方程的解 在一个中心力场方程，和克里斯托弗 函数与统计预测理论有关。 结果 多项式近似，这项工作的自然副产品， 预测理论在数字化处理中具有重要的应用价值。 电子信号。

Various topics in mathematical analysis associated with orthogonal polynomials will be taken up in this project. On the same interval, there are many classes of orthogonal polynomials. In each class, orthogonality is defined with respect to a measure (density) concentrated on the interval. The main theme of this work is a continuing effort to understand the relationship between the measure and the recurrence coefficients of the corresponding polynomials. Associated to each class of orthogonal polynomials is a unique algorithm which produces the higher order polynomials in terms of the lower. This recurrence is a simple linear relation, and it is the coefficients in the relation that are of considerable interest. Methods to be employed include minimizing polynomial integrals (the Christoffel function), asymptotic behavior of difference equations and eigenvalues of infinite tridiagonal matrices. The methods used for studying eigenvalues of matrices can be adapted to the study of solutions of the Schrodinger equation in a central field of force, and the Christoffel function is related to statistical prediction theory. Results of polynomial approximation, a natural by-product of this work, and prediction theory are valuable in digital processing of electronic signals.