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, which may be accelerated by one of a few different algorithms. Associated orthogonal polynomials arise from iterated Stieltjes transforms of orthogonal polynomials; hence, fast approximate conversion to classical cases combined with fast discrete sine and cosine transforms provides a modular mechanism for synthesis of singular integral transforms of classical orthogonal polynomial expansions.
Fast approximation of orthogonal matrices and application to PCA
Faculty of Automatic Control and Computers, University Politehnica Bucharest（布加勒斯特理工大学自动控制与计算机学院）；LCSL, Universitá di Genova（热那亚大学LCSL）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）
Fast and accurate computation of orthogonal moments for texture analysis
Department of Mathematics and Computer Science, University of Cagliari（卡利里大学数学与计算机科学系）；Department of Electrical and Electronic Engineering, University of Cagliari（卡利里大学电子与电气工程系）
