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Polynomial operators and local approximation of solutions of pseudo-differential equations on the sphere

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Abstract

We study the solutions of an equation of the form Lu=f, where L is a pseudo-differential operator defined for functions on the unit sphere embedded in a Euclidean space, f is a given function, and u is the desired solution. We give conditions under which the solution exists, and deduce local smoothness properties of u given corresponding local smoothness properties of f, measured by local Besov spaces. We study the global and local approximation properties of the spectral solutions, describe a method to obtain approximate solutions using values of f at points on the sphere and polynomial operators, and describe the global and local rates of approximation provided by our polynomial operators.

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References

  1. De Boor, C.: A practical guide to splines, Springer Verlag, New York, 1978

  2. Brown, G., Feng, D., Sheng, S.Y.: Kolmogorov widths of classes of smooth functions on the sphere S d-1. J. Complexity 18, 1001–1023 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. DeVore, R.A., Lorentz, G.G.: Constructive approximation, Springer Verlag, Berlin, 1993

  4. Driscoll, J.R., Healy, D.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. in Applied Math. 15, 202–250 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fasshauer, G.E., Schumaker, L.L.: Scattered Data Fitting on the Sphere. In: Mathematical Methods for Curves and Surfaces II (M. Dæhlen, T. Lyche, and L. L. Schumaker eds.), Vanderbilt Univ. Press, Nashville & London, 1998, pp. 117–166

  6. Freeden, W., Schreiner, M., Franke, R.: A Survey on Spherical Spline Approximation. Surveys Math. Indust. 7, 29–85 (1997)

    MATH  MathSciNet  Google Scholar 

  7. Freeden, W., Gervens, T., Schreiner, M.: Constructive approximation on the sphere, with applications to geomathematics. Clarendon Press, Oxford, 1998

  8. Gottlieb, D., Tadmor, E.: Recovering pointwise values of discontinuous data within spectral accuracy. In: Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of a 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston, 1985, pp. 357–375

  9. Grebenitcharsky, R.S., Sideris, M.G.: The compatibility conditions in altimetry–gravimetry boundary value problems. Journal of Geodesy 78, 626–636 (2005)

    Article  Google Scholar 

  10. Lizorkin, P.I., Rustamov, Kh.P.: Nikolskii–Besov spaces on the sphere in connection with approximation theory. Trudy Math. Inst. Steklov 204, 172–201 (1993) (Proceedings of the Steklov Institute of Mathematics 3, 149–172 (1994))

    MATH  MathSciNet  Google Scholar 

  11. Maier, T.: New components in geomagnetic field modelling wavelet-Mie-representation and wavelet-variances. Oberwolfach Reports Vol. 1, Report 27 European Mathematical Society, http://www.mfo.de, pp. 1417–1420

  12. Mhaskar, H.N.: Polynomial operators and local smoothness classes on the unit interval. J. Approx. Theory 131, 243–267 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Mhaskar, H.N.: On the representation of smooth functions on the sphere using finitely many bits. Adv. Comput. Harm. Anal. 18, Issue 3, May 2005, 215–233

  14. Mhaskar, H.N.: Weighted quadrature formulas and approximation by zonal function networks on the sphere. J. Complexity (In Press)

  15. Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comp. 70 no. 235, 1113–1130 (2001) (Corrigendum: Math. Comp. 71, 453–454 (2001))

    MathSciNet  Google Scholar 

  16. Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Approximation Theory, XI, Gatliburg, 2004 (C. K. Chui, M. Neamtu, L. L. Schumaker eds.), Nashboro Press, Brentwood, 2005, pp. 287–318

  17. Michel, D.: Mathematical methods in oceanography: approximation methods in ocean modeling. Oberwolfach Reports Vol. 1, Report 27 European Mathematical Society, http://www.mfo.de, pp. 1422–1425

  18. Müller, C.: Spherical harmonics. Lecture Notes in Mathematics, Vol. 17, Springer Verlag, Berlin, 1966

  19. Morton, T., Neamtu, M.: Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels. J. Approx. Theory 114, no. 2, 242–268 (2002)

    Google Scholar 

  20. Potts, D., Steidl, G., Tasche, M.: Fast algorithms for discrete polynomial transforms. Math. Comp. 67, 1577–1590 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Akademische Verlagsgesellschaft Geest& Portig, Leipzig, 1987

  22. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971

  23. Svensson, S.L.: Pseudo-differential operators – a new approach to the boundary value problems of physical geodesy. Manuscr. Geod. 8, 1–40 (1983)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Q. T. Le Gia

  2. Department of Mathematics, California State University Los Angeles, California, 90032, USA

    H. N. Mhaskar

Authors
  1. Q. T. Le Gia
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  2. H. N. Mhaskar
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Correspondence to H. N. Mhaskar.

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The research of this author was supported, in part, by grant DMS-0204704 from the National Science Foundation and grant W911NF-04-1-0339 from the U.S. Army Research Office

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Le Gia, Q., Mhaskar, H. Polynomial operators and local approximation of solutions of pseudo-differential equations on the sphere. Numer. Math. 103, 299–322 (2006). https://doi.org/10.1007/s00211-006-0676-z

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  • DOI: https://doi.org/10.1007/s00211-006-0676-z

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