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Subperiodic Trigonometric Hyperinterpolation

  • Gaspare Da Fies
  • Alvise Sommariva
  • Marco VianelloEmail author
Chapter

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.

Notes

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.

References

  1. 1.
    Adcock, B., Huybrechs, D.: On the resolution power of Fourier extensions for oscillatory functions. J. Comput. Appl. Math. 260, 312–336 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adcock, B., Platte, R.: A mapped polynomial method for high-accuracy approximations on arbitrary grids. SIAM J. Numer. Anal. 54, 2256–2281 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adcock, B., Ruan, J.: Parameter selection and numerical approximation properties of Fourier extensions from fixed data. J. Comput. Phys. 273, 453–471 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Adcock, B., Huybrechs, D., Vaquero, J.M.: On the numerical stability of Fourier extensions. Found. Comput. Math. 14, 635–687 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)CrossRefGoogle Scholar
  7. 7.
    Bos, L., Vianello, M.: Subperiodic trigonometric interpolation and quadrature. Appl. Math. Comput. 218, 10630–10638 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Curto, R.E., Fialkow, L.A.: A duality proof of Tchakaloff’s theorem. J. Math. Anal. Appl. 269, 519–532 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Da Fies, G.: Some results on subperiodic trigonometric approximation and quadrature. Master Thesis in Mathematics (advisor: Vianello, M.), University of Padova (2012)Google Scholar
  12. 12.
    Da Fies, G., Vianello, M.: Trigonometric Gaussian quadrature on subintervals of the period. Electron. Trans. Numer. Anal. 39, 102–112 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Da Fies, G., Vianello, M.: On the Lebesgue constant of subperiodic trigonometric interpolation. J. Approx. Theory 167, 59–64 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Da Fies, G., Vianello, M.: Product Gaussian quadrature on circular lunes. Numer. Math. Theory Methods Appl. 7, 251–264 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Da Fies, G., Sommariva, A., Vianello, M.: Algebraic cubature by linear blending of elliptical arcs. Appl. Numer. Math. 74, 49–61 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    De Marchi, S. Vianello, M., Xu, Y.: New cubature formulae and hyperinterpolation in three variables. BIT Numer. Math. 49, 55–73 (2009)Google Scholar
  17. 17.
    De Marchi, S., Sommariva, A., Vianello, M.: Multivariate Christoffel functions and hyperinterpolation. Dolomites Res. Notes Approx. DRNA 7, 26–33 (2014)Google Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ganesh, M., Mhaskar, H.N.: Matrix-free interpolation on the sphere. SIAM J. Numer. Anal. 44, 1314–1331 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9, 24–82 (1967)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004)zbMATHGoogle Scholar
  22. 22.
    Gautschi, W.: Orthogonal polynomials (in Matlab). J. Comput. Appl. Math. 178, 215–234 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gautschi, W.: Sub-range Jacobi polynomials. Numer. Algorithms 61, 649–657 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gentile, M., Sommariva, A., Vianello, M.: Polynomial approximation and quadrature on geographic rectangles. Appl. Math. Comput. 297, 159–179 (2017)MathSciNetGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefGoogle Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    Huybrechs, D.: On the Fourier extension of nonperiodic functions. SIAM J. Numer. Anal. 47, 4326–4355 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Leviatan, D., Sidon, J.: Monotone trigonometric approximation, Mediterr. J. Math. 12, 877–887 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Matthysen, R., Huybrechs, D.: Fast algorithms for the computation of Fourier extensions of arbitrary length. SIAM J. Sci. Comput. 38, A899–A922 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nevai, P.G.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 185 (1979)zbMATHGoogle Scholar
  32. 32.
    Piciocchi, V.: Subperiodic trigonometric hyperinterpolation in tensor-product spaces, Master Thesis in Mathematics (advisor: Vianello, M.), University of Padova (2014)Google Scholar
  33. 33.
    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)CrossRefGoogle Scholar
  34. 34.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)zbMATHGoogle Scholar
  35. 35.
    Sloan, I.H.: Interpolation and hyperinterpolation over general regions. J. Approx. Theory 83, 238–254 (1995)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Sommariva, A., Vianello, M.: Polynomial fitting and interpolation on circular sections. Appl. Math. Comput. 258, 410–424 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sommariva, A., Vianello, M.: Numerical hyperinterpolation over nonstandard planar regions. Math. Comput. Simul. 141, 110–120 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sommariva, A., Vianello, M.: HYPERTRIG: Matlab package for subperiodic trigonometric hyperinterpolation. Available online at: www.math.unipd.it/~marcov/subp.html
  39. 39.
    Tal-Ezer, H.: Nonperiodic trigonometric polynomial approximation. J. Sci. Comput. 60, 345–362 (2014)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Vianello, M.: Norming meshes by Bernstein-like inequalities. Math. Inequal. Appl. 17, 929–936 (2014)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Wade, J.: On hyperinterpolation on the unit ball. J. Math. Anal. Appl. 401, 140–145 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Gaspare Da Fies
    • 1
  • Alvise Sommariva
    • 2
  • Marco Vianello
    • 2
    Email author
  1. 1.Department of MathematicsAberystwyth UniversityWalesUK
  2. 2.Department of MathematicsUniversity of PadovaPadovaItaly

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