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Cardinal interpolation with general multiquadrics: convergence rates

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Abstract

This article pertains to interpolation of Sobolev functions at shrinking lattices \(h\mathbb {Z}^{d}\) from L p shift-invariant spaces associated with cardinal functions related to general multiquadrics, ϕ α, c (x) := (|x|2 + c 2)α. The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, L p error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h k) for functions with derivatives of order up to k in L p , \(1<p<\infty \)). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55 Courier Dover Publications (1972)

  2. Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43(4), 585–620 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baxter, B.J.C.: The asymptotic cardinal function of the multiquadratic \(\phi (r)=(r^{2}+ c^{2})^{\frac {1}{2}}\) as \(c\to \infty \). Comput. Math. Appl. 24(12), 1–6 (1992)

  4. de Boor, C., Ron, A.: Fourier analysis of approximation orders from principal shift-invariant spaces. Constr. Approx. 8(4), 427–462 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Boor, C., DeVore, R.A., Ron, A.: Approximation from shift-invariant subspaces of \(L_{2}(\mathbb {R}^{d})\). Trans. Amer. Math. Soc. 341(2), 787–806 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Buhmann, M.D.: Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics. Constr. Approx. 6(1), 21–34 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buhmann, M.D.: Multivariate cardinal interpolation with radial-basis functions. Constr. Approx. 6(3), 225–255 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, vol. 12. Cambridge University Press (2003)

  9. Buhmann, M.D., Dai, F.: Pointwise approximation with quasi-interpolation by radial basis functions. J. Approx. Theory 192, 156–192 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buhmann, M.D., Dyn, N.: Spectral convergence of multiquadric interpolation. Proc. Edinb. Math. Soc. (2) 36, 319–333 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buhmann, M.D., Michelli, C.A.: Multiquadric interpolation improved. Comput. Math. Appl. 24(2), 21–25 (1992)

    Article  MathSciNet  Google Scholar 

  12. Buhmann, M.D., Ron, A.: Radial base functions: L p –approximation orders with scattered centres. No. CMS-94-5 Wisconsin Univ-Madison center for mathematical sciences (1994)

  13. Driscoll, T., Fornberg, B.: Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43, 413–422 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fornberg, B., Wright, G.B.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5), 853–867 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fornberg, B., Flyer, N.: A primer on radial basis functions with applications to the geosciences. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 87. SIAM (2015)

  16. Friedlander, G., Joshi, M.: Introduction to the Theory of Distributions, 2nd edn. Cambridge University Press (1998)

  17. Hamm, K.: Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions. J. Approx. Theory 189, 101–122 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hamm, K., Ledford, J.: Cardinal interpolation with general multiquadrics. Adv. Comput. Math. 42(5), 1149–1186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hamm, K.: Nonuniform sampling and recovery of bandlimited functions in higher dimensions. J. Math. Anal Appl. 450(2), 1459–1478 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hangelbroek, T., Madych, W., Narcowich, F., Ward, J.: Cardinal interpolation with Gaussian kernels. J. Fourier Anal. Appl. 18, 67–86 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Holtz, O., Ron, A.: Approximation orders of shift-invariant subspaces of \({W_{2}^{s}}(\mathbb {R}^{d})\). J. Approx. Theory 132(1), 97–148 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jia, R.-Q.: Shift-invariant spaces on the real line. Proc. Amer. Math. Soc. 125(3), 785–793 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jia, R.-Q., Micchelli, C.A.: Using the refinement equation for the construction of prewavelets. II. Powers of two. In: Laurent, P.J., Le Méhauté, A., Schumaker, L.L. (eds.) Curves and Surfaces, pp. 209–246. Academic Press, New York (1991)

  24. Johnson, M.J.: On the approximation order of principal shift-invariant subspaces of \(L_{p}(\mathbb {R}^{d})\). J. Approx. Theory 91, 279–319 (1997)

    Article  MathSciNet  Google Scholar 

  25. Johnson, M.J.: Approximation in \(L_{p}(\mathbb {R}^{d})\) from spaces spanned by the perturbed integer translates of a radial function. J. Approx. Theory 107(2), 163–203 (2000)

    Article  MathSciNet  Google Scholar 

  26. Johnson, M.J.: Scattered data interpolation from principal shift-invariant spaces. J. Approx. Theory 113, 172–188 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kyriazis, G.C.: Approximation from shift-invariant spaces. Constr. Approx. 11(2), 141–164 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ledford, J.: Recovery of Paley-Wiener functions using scattered translates of regular interpolators. J. Approx. Theory 173, 1–13 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ledford, J.: Recovery of bivariate band-limited functions using scattered translates of the Poisson kernel. J. Approx. Theory 189, 170–180 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ledford, J.: On the convergence of regular families of cardinal interpolators. Adv. Comput. Math. 41, 357–371 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Madych, W.R.: Miscellaneous error bounds for multiquadric and related interpolators. Comput. Math. Appl. 24(12), 121–138 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Madych, W.R., Nelson, S.A.: Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70, 94–114 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Madych, W.R., Potter, E.H.: An estimate for multivariate interpolation. J. Approx. Theory 43, 132–139 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Translated from the Russian by W. J. A. Whyte. Translation Edited by I. N Sneddon. Pergamon Press, Oxford-New York-Paris (1965)

    MATH  Google Scholar 

  35. Riemenschneider, S.D., Sivakumar, N.: On the cardinal-interpolation operator associated with the one-dimensional multiquadric. East J. Approx. 7(4), 485–514 (1999)

    MathSciNet  MATH  Google Scholar 

  36. Riemenschneider, S.D., Sivakumar, N.: On cardinal interpolation by Gaussian radial-basis functions: properties of fundamental functions and estimates for Lebesgue constants. J. Anal. Math. 79, 33–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Riemenschneider, S.D., Sivakumar, N.: Gaussian radial basis functions: Cardinal interpolation of ℓ p and power growth data. Adv. Comput. Math. 11, 229–251 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Riemenschneider, S.D., Sivakumar, N.: Cardinal interpolation by Gaussian functions: a survey. J. Analysis 8, 157–178 (2000)

    MathSciNet  MATH  Google Scholar 

  39. Rudin, W.: Functional Analysis, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc, New York (1991)

    Google Scholar 

  40. Schlumprecht, T., Sivakumar, N.: On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian. J. Approx. Theory 159, 128–153 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Schoenberg, I.J.: Cardinal Spline Interpolation Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, vol. 12. Society for Industrial and Applied Mathematics, Philadelphia, Pa (1973)

    Google Scholar 

  42. Sivakumar, N.: A note on the Gaussian cardinal-interpolation operator. Proc. Edinb. Math. Soc. (2) 40, 137–150 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press (2005)

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Keaton Hamm

  2. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, USA

    Jeff Ledford

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  1. Keaton Hamm
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  2. Jeff Ledford
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Correspondence to Keaton Hamm.

Additional information

Communicated by: Karsten Urban

Part of the work for this article was completed when the first author was a graduate student at Texas A&M University, where he was partially supported by National Science Foundation grants DMS 1160633 and 1464713. The second author was partially supported by the Workshop in Analysis and Probability at Texas A&M University.

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Hamm, K., Ledford, J. Cardinal interpolation with general multiquadrics: convergence rates. Adv Comput Math 44, 1205–1233 (2018). https://doi.org/10.1007/s10444-017-9578-0

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  • DOI: https://doi.org/10.1007/s10444-017-9578-0

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