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On the norm of the hyperinterpolation operator on the \(d\)-dimensional cube. (English) Zbl 1362.65023

Summary: We obtain the asymptotic order of the operator norm of the hyperinterpolation operator on the cube \(I^d=[-1,1]^d\), \(d\geq 2\) with respect to the measure \(d\mu(x)=\prod_{i=1}^d\frac{dx_i}{\pi\sqrt{1-x_i^2}}\). This gives an affirmative answer to a conjecture raised by M. Caliari et al. [Comput. Math. Appl. 55, No. 11, 2490–2497 (2008; Zbl 1142.65312)].

MSC:

65D05 Numerical interpolation
65D32 Numerical quadrature and cubature formulas

Citations:

Zbl 1142.65312

Software:

Hyper2d; HyperCube
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Full Text: DOI

References:

[1] Sloan, I. H., Interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83, 238-254 (1995) · Zbl 0839.41006
[2] Caliari, M.; De Marchi, S.; Montagna, R.; Vianello, M., Hyper2d: a numerical code for hyperinterpolation on rectangles, Appl. Math. Comput., 183, 1138-1147 (2006) · Zbl 1105.65307
[3] Caliari, M.; De Marchi, S.; Vianello, M., Hyperinterpolation on the square, J. Comput. Appl. Math., 210, 1-2, 78-83 (2007) · Zbl 1151.65014
[4] Caliari, M.; De Marchi, S.; Vianello, M., Hyperinterpolation in the cube, Comput. Math. Appl., 55, 2490-2497 (2008) · Zbl 1142.65312
[5] Dai, Feng, On generalized hyperinterpolation on the sphere, Proc. Amer. Math. Soc., 134, 2931-2941 (2006) · Zbl 1108.41009
[6] Le Gia, Q. T.; Sloan, I. H., The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimension, Constr. Approx., 17, 2, 249-265 (2001) · Zbl 0989.41007
[7] Hansen, O.; Atkinson, K.; Chien, D., On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation, IMA J. Numer. Anal., 29, 2, 257-283 (2009) · Zbl 1163.65082
[8] Hesse, K.; Sloan, I. H., Hyperinterpolation on the sphere, (Govil, N. K.; Mhaskar, H. N.; Mohapatra, R. N.; Nashed, Z.; Szabados, J., Frontiers in Interpolation and Approximation (Dedicated to the Memory of Ambikeshwar Sharma) (2006), Chapman & Hall/CRC), 213-248 · Zbl 1194.41044
[9] De Marchi, S.; Vianello, M.; Xu, Y., New cubature formulas and hyperinterpolation in three variables, BIT, 49, 1, 55-73 (2009) · Zbl 1180.65031
[10] Reimer, M., Hyperinterpolation on the sphere at the minimal projection order, J. Approx. Theory, 104, 2, 272-286 (2000) · Zbl 0959.41001
[11] Reimer, M., Generalized hyperinterpolation on the sphere and the Newman-Shapiro operators, Constr. Approx., 18, 183-203 (2002) · Zbl 1002.41016
[12] Sloan, I. H.; Womersley, R. S., Constructive polynomial approximation on the sphere, J. Approx. Theory, 103, 1, 91-118 (2000) · Zbl 0946.41007
[13] Wade, J., On hyperinterpolation on the unit ball, J. Math. Anal. Appl., 401, 1, 140-145 (2013) · Zbl 1304.41003
[14] Wang, H., Optimal lower estimates for the worst case cubature error and the approximation by hyperinterpolation operators in the Sobolev space setting on the sphere, Int. J. Wavelets Multiresolut. Inf. Process., 7, 6, 813-823 (2009) · Zbl 1183.41033
[15] Szili, L.; Vertesi, P., On multivariate projection operators, J. Approx. Theory, 159, 154-164 (2009) · Zbl 1171.42001
[16] Xu, Y., Christoffel functions and Fourier series for multivariate orthogonal polynomials, J. Approx. Theory, 82, 2, 205-239 (1995) · Zbl 0874.42018
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