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A New Constructive and Elementary Proof of a Bernstein–Walsh Theorem, Improved to Infinite Order Convergence, for Functions C in an Intricate but Smooth Two-Dimensional Real Domain

  • John P. Boyd
Article

Abstract

We give a constructive proof that a C function f(x, y) can be approximated by a polynomial with an infinite order rate of convergence on a general two-dimensional domain that is specified as the set where a smooth function B(x, y) is non-negative and which can be embedded within a rectangle. We explicitly construct a C smoothed approximation to the characteristic function of the domain as \(\rho (x,y) \equiv \mathcal {H}([1 + SB(x,y)])\) where S > 0 is a constant, \(\mathcal {H}\) is a “ramp” (a smoothed approximation to the Heaviside step function) and ρ ≡ 1 on the domain Ω. The product f(x, y)ρ(x, y) is identically equal to f on the domain, but is of compact support. From this, we prove that ρf has a bivariate Chebyshev series on a rectangle that embeds Ω. We prove also that this expansion converges with increasing N, where N is the series truncation, faster than any finite inverse power of N, which is the definition of “infinite order” convergence. Bernstein–Walsh-type theorems have a long history, but the proofs use mathematical tools far removed from the education of the engineers and scientists. In contrast, our proof is constructive using tools accessible to applied practitioners. The proof of exponential convergence for functions with only C smoothness has not been previously given by any methodology.

Keywords

Bernstein–Walsh Stone–Weierstrass Chebyshev polynomial Spectral extension Fourier extension 

Mathematics Subject Classification (2010)

65N35 41A05 41A17 

Notes

Acknowledgements

This work was supported by the National Science Foundation through grant DMS 1521158. I thank the two referees for their careful and constructive comments.

References

  1. 1.
    Bagby, T., Bos, L., Levenberg, N.: Multivariate simultaneous approximation. Constr. Approx. 18, 569–577 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bagby, T., Levenberg, N.: Bernstein theorems. N. Z. J. Math. 22, 1–20 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bagby, T., Levenberg, N.: Bernstein theorems for elliptic equations. J. Approx. Theory 78, 190–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bernstein, S.N.: Sur l’ordre de la meilleure approximation des foncions continues per des polynomes de degré donné. Mémoires publiés par la class des sci. Acad. de Belgique 4, 1–103 (1912)Google Scholar
  5. 5.
    Bernstein, S.N.: Collected Works, Part I: Constructive Theory of Functions. Academy of Sciences USSR Press, Moscow (1952)Google Scholar
  6. 6.
    Bloom, T., Bos, L.P., Calvi, J. -P., Levenberg, N.: Polynomial interpolation and approximation in \(\mathbb {C}^{d}\). Ann. Polon. Math. 106, 53–81 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bochner, S., Martin, W.T.: Several Complex Variables. Princeton University Press, Princeton (1948)zbMATHGoogle Scholar
  8. 8.
    Bos, L., Calvi, J. -P., Levenberg, N., Sommariva, A., Vianello, M.: Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points. Math. Comput. 80(280), 1623–1638 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boyd, J.P.: Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint. Appl. Math. Lett. 9, 11–15 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications, Mineola (2001)zbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boyd, J.P.: Asymptotic Fourier coefficients for a C bell (smoothed-“top-hat” function) and the Fourier extension problem. J. Sci. Comput. 29, 1–24 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dyn, N., Floater, M.S.: Multivariate polynomial interpolation on lower sets. J. Approx. Theory 177, 34–42 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)CrossRefzbMATHGoogle Scholar
  15. 15.
    Höllig, K.: Finite Element Methods with B-Splines Frontiers in Applied Mathematics, vol. 26. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  16. 16.
    Kraus, C.: Multivariate polynomial approximation of powers of the euclidean distance function. Constr. Approx. 27, 323–327 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kraus, C.: Bernstein–Walsh type theorems for real analytic functions in several variables. Constr. Approx. 33, 191–217 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Leaf, G.K., Kaper, H.G.: L -error bounds for multivariate Lagrange approximation. SIAM J. Numer. Anal. 11, 363–381 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Levenberg, N.: Approximation in \(\mathbb {C}^{N}\). Surv. Approx. Theory 2, 92–140 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Li, S., Boyd, J.P.: Spectral methods in non-tensor geometry, Part II: Chebyshev versus Zernike polynomials, gridding strategies and spectral extension on squircle-bounded and perturbed-quadrifolium domains. Appl. Math. Comput. 269, 759–774 (2015)MathSciNetGoogle Scholar
  21. 21.
    Li, S., Boyd, J.P.: Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids. J. Comput. Phys. 281, 653–668 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li, S., Boyd, J.P.: Symmetrizing grids, radial basis functions, and Chebyshev and Zernike polynomials for the D 4 symmetry group; Interpolation within a squircle, Part I. J. Comput. Phys. 258, 931–947 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lyness, J.N.: The calculation of trigonometric Fourier coefficients. J. Comput. Phys. 54, 57–73 (1984)CrossRefzbMATHGoogle Scholar
  24. 24.
    Narayan, A., Xiu, D.: Constructing nested nodal sets for multivariate polynomial interpolation. SIAM J. Sci. Comput. 35, A2293–A2315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pawłucki, W., Pleśniak, W.: Markov’s inequality and C functions on sets with polynomial cusps. Math. Ann. 275, 467–480 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pierzchała, R.: On the Bernstein–Walsh–Siciak theorem. Stud. Math. 212, 55–63 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pierzchała, R.: Approximation of holomorphic functions on compact subsets of \(\mathbb {R}^{N}\). Constr. Approx. 41, 133–155 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pierzchała, R: An estimate for the Siciak extremal function – subanalytic geometry approach. J. Math. Anal. Appl. 430, 755–776 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pleśniak, W.: Multivariate Jackson inequality. J. Comput. Appl. Math. 233, 815–820 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sapogov, N.A.: Uber die beste Annäherung analytischer Funktionen von mehreren Veränderlichen und über Reihen von Polynomen. Mat. Sb. 38, 331–336 (1956). In RussianMathSciNetzbMATHGoogle Scholar
  31. 31.
    Schultz, M.H.: L -multivariate approximation theory. SIAM J. Numer. Anal. 6, 161–183 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sederberg, T.W., Goldman, R.N.: Algebraic geometry for computer-aided geometric design. IEEE Comput. Graph. Appl. 6, 52–59 (1986)CrossRefGoogle Scholar
  33. 33.
    Shen, J., Yu, H.: Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems. SIAM J. Sci. Comput. 32, 3228–3250 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Siciak, J.: On some extremal functions and their applications in the theory of analytic functions of several complex variables. Trans. Am. Math. Soc. 105, 322–357 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Walsh, J.L., Russell, H.G.: On the convergence and overconvergence of sequences of polynomials of best simultaneous approximation to several functions analytic in distinct regions. Trans. Am. Math. Soc. 36, 13–28 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Climate and Space Science and EngineeringUniversity of MichiganAnn ArborUSA

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