Minimal Cubature rules and polynomial interpolation in two variables

Abstract

Minimal cubature rules of degree $4n-1$ for the weight functions $$ W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1{]}^2$ are constructed explicitly and are shown to be closed related to the Gaussian cubature rules in a domain bounded by two lines and a parabola. Lagrange interpolation polynomials on the nodes of these cubature rules are constructed and their Lebesgue constants are determined.

23 pages

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