Computation of connection coefficients and measure modifications for orthogonal polynomials

We observe that polynomial measure modifications for families of univariate orthogonal polynomials imply sparse connection coefficient relations. We therefore propose connecting L ² expansion coefficients between a polynomial family and a modified family by a sparse transformation. Accuracy and conditioning of the connection and its inverse are explored. The connection and recurrence coefficients can simultaneously be obtained as the Cholesky decomposition of a matrix polynomial involving the Jacobi matrix; this property extends to continuous, non-polynomial measure modifications on finite intervals. We conclude with an example of a useful application to families of Jacobi polynomials with parameters (γ,δ) where the fast Fourier transform may be applied in order to obtain expansion coefficients whenever 2γ and 2δ are odd integers.