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

Abstract

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861–877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal O}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal O}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> computational scheme for barycentric weights.