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Riemann Localisation on the Sphere

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Abstract

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.

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Notes

  1. Let R(t) be a rational polynomial taking the form \(R(t)=p(t)/q(t)\), where p(t) and q(t) are polynomials with \(q\ne 0\). The degree of R(t) is \(\deg (R):=\deg (p)-\deg (q)\).

References

  1. Antoine, J.-P., Vandergheynst, P.: Wavelets on the two-sphere and other conic sections. J. Fourier Anal. Appl. 13(4), 369–386 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Adaptive density estimation for directional data using needlets. Ann. Stat. 37(6A), 3362–3395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Asymptotics for spherical needlets. Ann. Stat. 37(3), 1150–1171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belinsky, E., Dai, F., Ditzian, Z.: Multivariate approximating averages. J. Approx. Theory 125(1), 85–105 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publ. Res. Inst. Math. Sci. Ser. A 4, 201–268 (1968/1969)

  6. Bonami, A., Clerc, J.-L.: Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques. Trans. Am. Math. Soc. 183, 223–263 (1973)

    MATH  Google Scholar 

  7. Brandolini, L., Colzani, L.: Localization and convergence of eigenfunction expansions. J. Fourier Anal. Appl. 5(5), 431–447 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carbery, A., Soria, F.: Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an \(L^2\)-localisation principle. Rev. Mat. Iberoamericana 4(2), 319–337 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carbery, A., Soria, F.: Pointwise Fourier inversion and localisation in \({\mathbb{R}^n}\). J. Fourier Anal. Appl. 3(special issue), 847–858 (1997)

  10. Carbery, A., Soria, F.: Sets of divergence for the localization problem for Fourier integrals. C. R. Acad. Sci. Paris Sér. I Math. 325(12), 1283–1286 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/. Release 1.0.9 of 2014-08-29. Online companion to [21]

  12. Freeden, W., Mayer, C.: Wavelets generated by layer potentials. Appl. Comput. Harmon. Anal. 14(3), 195–237 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Freeden, W., Windheuser, U.: Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl. Comput. Harmon. Anal. 4(1), 1–37 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frenzen, C.L., Wong, R.: A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Can. J. Math. 37(5), 979–1007 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gerhards, C.: A combination of downward continuation and local approximation for harmonic potentials. Inverse Prob. 30(8), 085004, 30 (2014)

  16. Hille, E., Klein, G.: Riemann’s localization theorem for Fourier series. Duke Math. J. 21, 587–591 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ivanov, K., Petrushev, P., Xu, Y.: Sub-exponentially localized kernels and frames induced by orthogonal expansions. Math. Z. 264(2), 361–397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marinucci, D., Peccati, G.: Random Fields on the Sphere. Representation, Limit Theorems and Cosmological Applications. London Mathematical Society Lecture Note Series, vol. 389. Cambridge University Press, Cambridge (2011)

  19. Mhaskar, H.N.: On the representation of smooth functions on the sphere using finitely many bits. Appl. Comput. Harmon. Anal. 18(3), 215–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38(2), 574–594 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Petrushev, P., Xu, Y.: Localized polynomial frames on the interval with Jacobi weights. J. Fourier Anal. Appl. 11(5), 557–575 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pinsky, M.A.: Pointwise Fourier inversion and related eigenfunction expansions. Commun. Pure Appl. Math. 47(5), 653–681 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pinsky, M.A.: Pointwise Fourier inversion in several variables. Not. Am. Math. Soc. 42(3), 330–334 (1995)

    MathSciNet  MATH  Google Scholar 

  24. Pinsky, M.A., Taylor, M.E.: Pointwise Fourier inversion: a wave equation approach. J. Fourier Anal. Appl. 3(6), 647–703 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral concentration on a sphere. SIAM Rev. 48(3), 504–536 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stein, E.M., Shakarchi, R.: Fourier Analysis: An Introduction. Princeton Lectures in Analysis, vol. 1. Princeton University Press, Princeton (2003)

  27. Szegő, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. 23, 4th edn. AMS, Providence (2003)

  28. Taylor, M.E.: Pointwise Fourier inversion on tori and other compact manifolds. J. Fourier Anal. Appl. 5(5), 449–463 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Taylor, M.E.: Eigenfunction expansions and the Pinsky phenomenon on compact manifolds. J. Fourier Anal. Appl. 7(5), 507–522 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Taylor, M.E.: The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions. Commun. Partial Differ. Equ. 27(3–4), 565–605 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Telyakovskiĭ, S.A.: The Riemann localization principle and an estimate for the rate of convergence. Sovrem. Mat. Fundam. Napravl. 25, 178–181 (2007)

    Google Scholar 

  32. Wang, K., Li, L.: Harmonic analysis and approximation on the unit sphere. Science Press, Beijing (2006)

    Google Scholar 

  33. Wang, Y.G., Le Gia, Q.T., Sloan, I.H., Womersley, R.S.: Fully discrete needlet approximation on the sphere. Appl. Comput. Harmon. Anal. (2016) (in press)

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Acknowledgments

The authors would like to thank Christian Gerhards and Leonardo Colzani for their discussion and comments on the convergence of the Fourier local convolution and the localisation principle. The authors also thank the anonymous referees for their comments on simplifying the proof of Theorem 3.2. This research was supported under the Australian Research Council’s Discovery Project DP120101816. The first author was supported under the University International Postgraduate Award (UIPA) of UNSW Australia.

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  1. Yu Guang Wang

    Present address: Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Kensington, Australia

    Yu Guang Wang, Ian H. Sloan & Robert S. Womersley

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  1. Yu Guang Wang
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  2. Ian H. Sloan
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Correspondence to Yu Guang Wang.

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Communicated by Pencho Petrushev.

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Wang, Y.G., Sloan, I.H. & Womersley, R.S. Riemann Localisation on the Sphere. J Fourier Anal Appl 24, 141–183 (2018). https://doi.org/10.1007/s00041-016-9496-4

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