It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial measure modification problem has such a displacement structure, unlocking a collection of fast algorithms for computing connection coefficients (as the upper-triangular Cholesky factor) between a known orthogonal polynomial family and the modified family. In general, the ${\cal O}(n^3)$ complexity is reduced to ${\cal O}(n^2)$, and if the symmetric Gram matrix has upper and lower bandwidth b, then the ${\cal O}(b^2n)$ complexity for a banded Cholesky factorization is reduced to ${\cal O}(b n)$. In the case of modified Chebyshev polynomials, we show that the Gram matrix is a symmetric Toeplitz-plus-Hankel matrix, and if the modified Chebyshev moments decay algebraically, then a hierarchical off-diagonal low-rank structure is observed in the Gram matrix, enabling a further reduction in the complexity of an approximate Cholesky factorization powered by randomized numerical linear algebra.

翻译：众所周知，具有低海森伯格结构位移秩的矩阵在特定矩阵分解方面享有快速算法。我们证明了正交多项式测度修正问题中格拉姆矩阵的$n\times n$主有限截断具有此类位移结构，从而解锁了一组用于计算已知正交多项式族与修正族之间连接系数（作为上三角Cholesky因子）的快速算法。一般而言，计算复杂度从${\cal O}(n^3)$降低至${\cal O}(n^2)$；若对称格拉姆矩阵具有上下带宽$b$，则带状Cholesky分解的${\cal O}(b^2n)$复杂度可进一步降至${\cal O}(b n)$。对于修正切比雪夫多项式的情形，我们证明其格拉姆矩阵为对称Toeplitz-plus-Hankel矩阵，且当修正切比雪夫矩呈代数衰减时，格拉姆矩阵中会呈现层次化的非对角低秩结构，从而借助随机数值线性代数方法实现近似Cholesky分解复杂度的进一步降低。