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Hyperinterpolation in the cube. (English) Zbl 1142.65312

Summary: We construct an hyperinterpolation formula of degree \(n\) in the three-dimensional cube, by using the numerical cubature formula for the product Chebyshev measure given by the product of a (near) minimal formula in the square with Gauss-Chebyshev-Lobatto quadrature. The underlying function is sampled at \(N\sim n^{3}/2\) points, whereas the hyperinterpolation polynomial is determined by its \((n+1)(n+2)(n+3)/6\sim n^{3}/6\) coefficients in the trivariate Chebyshev orthogonal basis. The effectiveness of the method is shown by a numerical study of the Lebesgue constant, which turns out to increase like \(\log ^{3}(n)\), and by the application to several test functions.

MSC:

65D05 Numerical interpolation
65D30 Numerical integration
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