We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.

翻译：我们描述了一种快速算法，用于近似计算一族正交多项式与另一族具有多项式或有理修正测度的正交多项式之间的连接系数。该连接系数通过无限维带状矩阵分解计算，且可在截断阶数的线性复杂度内获得修正的雅可比矩阵。通过构建具有修正经典权函数的正交多项式族，使其支持带状微分矩阵，从而实现了基于修正经典正交多项式的稀疏谱方法。