Minimal cubature rules and polynomial interpolation in two variables

Abstract

Minimal cubature rules of degree 4n − 1 for the weight functionson [−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.

Optimal Points for Cubature Rules and Polynomial Interpolation on a Square

“…In the special cases of α = β = − 1 2 and γ = ± 1 2 , these are exactly the Chebyshev weight functions. It was proved recently in [29,31], rather surprisingly, that the results in the previous section can be extended to these weight functions. First, however, we describe a family of mutually orthogonal polynomials.…”

“…Let L α,β n f be the interpolation polynomial based on X α,β 2m when n = 2m and on X α,β 2m+1 when n = 2m + 1, as defined in (7). The asymptotics of the Lebesgue constants for these interpolation polynomials can be determined [29,31].…”

Multivariate polynomial interpolation on Lissajous–Chebyshev nodes

“…In [16], this statement is extended to polynomial spaces of Chebyshev polynomials of the second, third and fourth kind. Using the zeros of Jacobi polynomials instead of the zeros of Chebyshev polynomials a corresponding extension can be found in [24].…”

Minimal cubature rules and polynomial interpolation in two variables II

“…The nodes of these cubature rules are common zeros of certain orthogonal polynomials of degree n and, in the case of the second family, one quasi-orthogonal polynomial of degree n+1 that does not belong to the idea generated by those orthogonal polynomials of degree n. The second family of cubature rules are explicitly constructed because they are related to the product Gauss-Radau cubature rules with respect to the product Jacobi weights. For all practical considerations, the second family is better and their study resembles the case of n = 2m in [17]. In addition, we will also give explicit formulas of the Lagrange interpolation polynomials based on the nodes of the near minimal cubature rules.…”

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As a complement to [17], minimal cubature rules of degree 4m + 1 for the weight functionson [−1, 1] 2 are shown to exist and near minimal cubature rules of the same degree with one node more than minimal are constructed explicitly. The Lagrange interpolation polynomials on the nodes of the near minimal cubature rules are also studied.

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