Hyperinterpolation on the square

Authors:
Marco Caliari;Stefano De Marchi;Marco Vianello
Affiliations:
Department of Pure and Applied Mathematics, University of Padova, Via Trieste 63, 35121 Padova, Italy;Department of Computer Science, University of Verona, Italy;Department of Pure and Applied Mathematics, University of Padova, Via Trieste 63, 35121 Padova, Italy
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

Citing 9
Cited 2

Minimal cubature formulae of degree 2k - 1 for two classical functions

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On multivariate Lagrange interpolation

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Christoffel functions and Fourier series for multivariate orthogonal polynomials

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Polynomial interpolation and hyperinterpolation over general regions

Journal of Approximation Theory
Lagrange interpolation on Chebyshev points of two variables

Journal of Approximation Theory
Constructive polynomial approximation on the sphere

Journal of Approximation Theory
Cubature formulae and orthogonal polynomials

Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. V: quadrature and orthogonal polynomials
A Numerical Study of the Xu Polynomial Interpolation Formula in Two Variables

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On the Lebesgue constant for the Xu interpolation formula

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Hyperinterpolation in the cube

Computers & Mathematics with Applications
Computing Fekete and Lebesgue points: Simplex, square, disk

Journal of Computational and Applied Mathematics

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Abstract

We show that hyperinterpolation at (near) minimal cubature points for the product Chebyshev measure, along with Xu compact formula for the corresponding reproducing kernel, provide a simple and powerful polynomial approximation formula in the uniform norm on the square. The Lebesgue constant of the hyperinterpolation operator grows like log^2 of the degree, as that of quasi-optimal interpolation sets recently proposed in the literature. Moreover, we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points, and we compare it with interpolation formulas at the same set of points.