Abstract
Using recent results on subperiodic trigonometric Gaussian quadrature and the construction of subperiodic trigonometric orthogonal bases, we extend Sloan’s notion of hyperinterpolation to trigonometric spaces on subintervals of the period. The result is relevant, for example, to function approximation on spherical or toroidal rectangles.
Dedicated to Ian H. Sloan on the occasion of his 80th birthday.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Adcock, B., Huybrechs, D.: On the resolution power of Fourier extensions for oscillatory functions. J. Comput. Appl. Math. 260, 312–336 (2014)
Adcock, B., Platte, R.: A mapped polynomial method for high-accuracy approximations on arbitrary grids. SIAM J. Numer. Anal. 54, 2256–2281 (2016)
Adcock, B., Ruan, J.: Parameter selection and numerical approximation properties of Fourier extensions from fixed data. J. Comput. Phys. 273, 453–471 (2014)
Adcock, B., Huybrechs, D., Vaquero, J.M.: On the numerical stability of Fourier extensions. Found. Comput. Math. 14, 635–687 (2014)
Berschneider, G., Sasvri, Z.: On a theorem of Karhunen and related moment problems and quadrature formulae, Spectral theory, mathematical system theory, evolution equations, differential and difference equations. Oper. Theory Adv. Appl. 221, 173–187 (2012)
Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)
Bos, L., Vianello, M.: Subperiodic trigonometric interpolation and quadrature. Appl. Math. Comput. 218, 10630–10638 (2012)
Boyd, J.P.: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178, 118–160 (2002)
Bruno, O.P., Han, Y., Pohlman, M.M.: Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis. J. Comput. Phys. 227, 1094–1125 (2007)
Curto, R.E., Fialkow, L.A.: A duality proof of Tchakaloff’s theorem. J. Math. Anal. Appl. 269, 519–532 (2002)
Da Fies, G.: Some results on subperiodic trigonometric approximation and quadrature. Master Thesis in Mathematics (advisor: Vianello, M.), University of Padova (2012)
Da Fies, G., Vianello, M.: Trigonometric Gaussian quadrature on subintervals of the period. Electron. Trans. Numer. Anal. 39, 102–112 (2012)
Da Fies, G., Vianello, M.: On the Lebesgue constant of subperiodic trigonometric interpolation. J. Approx. Theory 167, 59–64 (2013)
Da Fies, G., Vianello, M.: Product Gaussian quadrature on circular lunes. Numer. Math. Theory Methods Appl. 7, 251–264 (2014)
Da Fies, G., Sommariva, A., Vianello, M.: Algebraic cubature by linear blending of elliptical arcs. Appl. Numer. Math. 74, 49–61 (2013)
De Marchi, S. Vianello, M., Xu, Y.: New cubature formulae and hyperinterpolation in three variables. BIT Numer. Math. 49, 55–73 (2009)
De Marchi, S., Sommariva, A., Vianello, M.: Multivariate Christoffel functions and hyperinterpolation. Dolomites Res. Notes Approx. DRNA 7, 26–33 (2014)
Dominguez, V., Ganesh, M.: Interpolation and cubature approximations and analysis for a class of wideband integrals on the sphere. Adv. Comput. Math. 39, 547–584 (2013)
Ganesh, M., Mhaskar, H.N.: Matrix-free interpolation on the sphere. SIAM J. Numer. Anal. 44, 1314–1331 (2006)
Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9, 24–82 (1967)
Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004)
Gautschi, W.: Orthogonal polynomials (in Matlab). J. Comput. Appl. Math. 178, 215–234 (2005)
Gautschi, W.: Sub-range Jacobi polynomials. Numer. Algorithms 61, 649–657 (2012)
Gentile, M., Sommariva, A., Vianello, M.: Polynomial approximation and quadrature on geographic rectangles. Appl. Math. Comput. 297, 159–179 (2017)
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, 257–283 (2009)
Hesse, K., Sloan, I.H.: Hyperinterpolation on the sphere. In: Frontiers in Interpolation and Approximation. Pure and Applied Mathematics, vol. 282, pp. 213–248. Chapman and Hall/CRC, Boca Raton, FL (2007)
Huybrechs, D.: On the Fourier extension of nonperiodic functions. SIAM J. Numer. Anal. 47, 4326–4355 (2014)
Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an O(N −1) time step restriction. J. Comput. Phys. 104, 457–469 (1993)
Leviatan, D., Sidon, J.: Monotone trigonometric approximation, Mediterr. J. Math. 12, 877–887 (2015)
Matthysen, R., Huybrechs, D.: Fast algorithms for the computation of Fourier extensions of arbitrary length. SIAM J. Sci. Comput. 38, A899–A922 (2016)
Nevai, P.G.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 185 (1979)
Piciocchi, V.: Subperiodic trigonometric hyperinterpolation in tensor-product spaces, Master Thesis in Mathematics (advisor: Vianello, M.), University of Padova (2014)
Piessens, R.: Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms. In: Numerical Integration (Halifax, N.S., 1986). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 203, pp. 35–51. Reidel, Dordrecht (1987)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Sloan, I.H.: Interpolation and hyperinterpolation over general regions. J. Approx. Theory 83, 238–254 (1995)
Sommariva, A., Vianello, M.: Polynomial fitting and interpolation on circular sections. Appl. Math. Comput. 258, 410–424 (2015)
Sommariva, A., Vianello, M.: Numerical hyperinterpolation over nonstandard planar regions. Math. Comput. Simul. 141, 110–120 (2017)
Sommariva, A., Vianello, M.: HYPERTRIG: Matlab package for subperiodic trigonometric hyperinterpolation. Available online at: www.math.unipd.it/~marcov/subp.html
Tal-Ezer, H.: Nonperiodic trigonometric polynomial approximation. J. Sci. Comput. 60, 345–362 (2014)
Vianello, M.: Norming meshes by Bernstein-like inequalities. Math. Inequal. Appl. 17, 929–936 (2014)
Wade, J.: On hyperinterpolation on the unit ball. J. Math. Anal. Appl. 401, 140–145 (2013)
Wang, H., Wang, K., Wang, X.: On the norm of the hyperinterpolation operator on the d-dimensional cube. Comput. Math. Appl. 68, 632–638 (2014)
Acknowledgements
Supported by the Horizon 2020 ERA-PLANET European project “GEOEssential”, by the EU project H2020-MSCA-RISE-2014-644175-MATRIXASSAY, by the DOR funds and by the biennial projects CPDA143275 and BIRD163015 of the University of Padova, and by the GNCS-INdAM.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Fies, G.D., Sommariva, A., Vianello, M. (2018). Subperiodic Trigonometric Hyperinterpolation. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-72456-0_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72455-3
Online ISBN: 978-3-319-72456-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)