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Barycentric interpolation is arguably the method of choice for numericalpolynomial interpolation. The polynomial interpolant is expressed in terms offunction values using the so-called barycentric weights, which depend on theinterpolation points. Few explicit formulae for these barycentric weights areknown. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authorshave shown that the barycentric weights of the roots of Legendre polynomialscan be expressed explicitly in terms of the weights of the correspondingGaussian quadrature rule. This idea was subsequently implemented in the Chebfunpackage [L. N. Trefethen and others, The Chebfun Development Team, 2011] and inthe process generalized by the Chebfun authors to the roots of Jacobi, Laguerreand Hermite polynomials. In this paper, we explore the generality of the linkbetween barycentric weights and Gaussian quadrature and show that suchrelationships are related to the existence of lowering operators for orthogonalpolynomials. We supply an exhaustive list of cases, in which all known formulaeare recovered and also some new formulae are derived, including the barycentricweights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcalO}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale andTownsend, this leads to an ${\mathcal O}(n)$ computational scheme forbarycentric weights.

Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials