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Poisson Wavelets on \(n\)-Dimensional Spheres

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Abstract

In this paper, Poisson wavelets on \(n\)-dimensional spheres, derived from Poisson kernel, are introduced and characterized. We compute their Gegenbauer expansion with respect to the origin of the sphere, as well as with respect to the field source. Further, we give recursive formulae for their explicit representations and we show how the wavelets are localized in space. Also their Euclidean limit is calculated explicitly and its space localization is described. We show that Poisson wavelets can be treated as wavelets derived from approximate identities and we give two inversion formulae.

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References

  1. Antoine, J.-P.: Wavelets and Wavelet Frames on the 2-Sphere. Contemporary Problems in Mathematical Physics, pp. 344–362. World Science Publisher, Hackensack (2006)

    Google Scholar 

  2. Antoine, J.-P., Demanet, L., Jacques, L., Vandergheynst, P.: Wavelets on the sphere: implemantation and approximations. Appl. Comput. Harmon. Anal. 13(3), 177–200 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Antoine, J.-P., Murenzi, R., Vandergheynst, P., Ali, S.T.: Two-Dimensional Wavelets and Their Relatives. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  4. Antoine, J.-P., Vandergheynst, P.: Wavelets on the n-sphere and related manifolds. J. Math. Phys. 39(8), 3987–4008 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Antoine, J.-P., Vandergheynst, P.: Wavelets on the 2-sphere: a group-theoretical approach. Appl. Comput. Harmon. Anal. 7(3), 262–291 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bernstein, S.: Spherical singular integrals, monogenic kernels and wavelets on the three-dimensional sphere. Adv. Appl. Clifford Algebr. 19(2), 173–189 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bernstein, S., Ebert, S.: Kernel based wavelets on \(S^3\). J. Concr. Appl. Math. 8(1), 110–124 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Bernstein, S., Ebert, S.: Wavelets on \(S^3\) and \(SO(3)\)—their construction, relation to each other and Radon transform of wavelets on \(SO(3)\). Math. Methods Appl. Sci. 33(16), 1895–1909 (2010)

    MATH  MathSciNet  Google Scholar 

  10. Bogdanova, I., Vandergheynst, P., Antoine, J.-P., Jacques, L., Morvidone, M.: Stereographic wavelet frames on the sphere. Appl. Comput. Harmon. Anal. 19(2), 223–252 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cerejeiras, P., Ferreira, M., Kähler, U.: Clifford analysis and the continuous spherical wavelet transform. In: Wavelet Analysis and Applications, pp. 173–184. Appl. Numer. Harmon. Anal., Birkhuser, Basel (2007)

  12. Chambodut, A., Panet, I., Mandea, M., Diament, M., Holschneider, M., Jamet, O.: Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163, 875–899 (2005)

    Article  Google Scholar 

  13. Ebert, S.: Wavelets on Lie groups and homogeneous spaces. Ph.D. thesis, Freiberg (2011)

  14. Ebert, S., Bernstein, S., Cerejeiras, P., Káhler, U.: Nonzonal wavelets on \({\cal {S}}^{N}\). In: 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, 2009

  15. Ferreira, M.: Spherical continuous wavelet transforms arising from sections of the Lorentz group. Appl. Comput. Harmon. Anal. 26(2), 212–229 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere. With Applications to Geomathematics, Numerical Mathematics and Scientific Computation. Clarendon, Oxford University Press, New York (1998)

    MATH  Google Scholar 

  17. Freeden, W., Schreiner, M.: Non-orthogonal expansions on the sphere. Math. Methods Appl. Sci. 18(2), 83–120 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Freeden, W., Schreiner, M.: Orthogonal and nonorthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constr. Approx. 14(4), 493–515 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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  MATH  MathSciNet  Google Scholar 

  20. Freeden, W., Windheuser, U.: Spherical wavelet transform and its discretization. Adv. Comput. Math. 5(1), 51–94 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Geller, D., Lan, X., Marinucci, D.: Spin needlets spectral estimation. Electron. J. Stat. 3, 1497–1530 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Geller, D., Marinucci, D.: Spin wavelets on the sphere. J. Fourier Anal. Appl. 16(6), 840–884 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Geller, D., Mayeli, A.: Besov spaces and frames on compact manifolds. Indiana Univ. Math. J. 58(5), 2003–2042 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Geller, D., Mayeli, A.: Continuous wavelets on compact manifolds. Math. Z. 262(4), 895–927 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Geller, D., Mayeli, A.: Nearly tight frames and space-frequency analysis on compact manifolds. Math. Z. 263(2), 235–264 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Geller, D., Mayeli, A.: Nearly tight frames of spin wavelets on the sphere. Sampl. Theory Signal Image Process. 9(1–3), 25–57 (2010)

    MATH  MathSciNet  Google Scholar 

  27. Geller, D., Pesenson, I.Z.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21(2), 334–371 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier/Academic Press, Amsterdam (2007)

    MATH  Google Scholar 

  29. Hayn, M., Holschneider, M.: Directional spherical multipole wavelets. J. Math. Phys. 50(7), 073512, 11pp (2009)

  30. Holschneider, M.: Continuous wavelet transforms on the sphere. J. Math. Phys. 37(8), 4156–4165 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Holschneider, M., Chambodut, A., Mandea, M.: From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys. Earth Planet. Inter. 135, 107–123 (2003)

    Article  Google Scholar 

  32. Holschneider, M., Iglewska-Nowak, I.: Poisson wavelets on the sphere. J. Fourier Anal. Appl. 13, 405–419 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Iglewska-Nowak, I.: Continuous wavelet transforms on n-dimensional spheres. Appl. Comput. Harmon. Anal. http://dx.doi.org/10.1016/j.acha.2014.09.006 (2014)

  34. Iglewska-Nowak, I.: Poisson wavelet frames on the sphere. Ph.D. thesis, Potsdam (2007)

  35. Iglewska-Nowak, I.: Semi-continuous and discrete wavelet frames on n-dimensional spheres, preprint, https://www.researchgate.net/profile/Ilona_Iglewska-Nowak/publications. Accessed 06 Oct 2014

  36. Iglewska-Nowak, I., Holschneider, M.: Frames of Poisson wavelets on the sphere. Appl. Comput. Harmon. Anal. 28, 227–248 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lan, X., Marinucci, D.: On the dependence structure of wavelet coefficients for spherical random fields. Stoch. Process. Appl. 119(10), 3749–3766 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mayeli, A.: Asymptotic uncorrelation for Mexican needlets. J. Math. Anal. Appl. 363(1), 336–344 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  39. McEwen, J.D., Hobson, M.P., Mortlock, D.J., Lasenby, A.N.: Fast directional continuous spherical wavelet transform algorithm. IEEE Trans. Signal Process. 55(2), 520–529 (2007)

    Article  MathSciNet  Google Scholar 

  40. McEwen, J.D., Vandergheynst, P., Wiaux, Y.: On the computation of directional scale-discretized wavelet transforms on the sphere. doi:10.1117/12.2022889

  41. McEwen, J.D., Wiaux, Y.: A novel sampling theorem on the sphere. IEEE Trans. Signal Process. 59(12), 5876–5887 (2011)

    Article  MathSciNet  Google Scholar 

  42. Narcowich, F.J., Petrushev, P., Ward, J.D.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, no. 32. Princeton University Press, Princeton, NJ (1971)

    MATH  Google Scholar 

  45. Vilenkin, NJa: Special functions and the theory of group representations. Translations of Mathematical Monographs, vol. 22. American Mathematical Society, Providence (1968)

    Google Scholar 

  46. Wiaux, Y., Jacques, L., Vandergheynst, P.: Correspondence principle between spherical nad Euclidean wavelets. Astrophys. J. 632(1), 15–28 (2005)

    Article  Google Scholar 

  47. Wiaux, Y., McEwen, J.D., Vielva, P.: Complex data processing: fast wavelet analysis on the sphere. J. Fourier Anal. Appl. 13(4), 477–493 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, West Pomeranian University of Technology, al. Piastów 17, 70–310, Szczecin, Poland

    Ilona Iglewska-Nowak

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  1. Ilona Iglewska-Nowak
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Correspondence to Ilona Iglewska-Nowak.

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Communicated by Hans G. Feichtinger.

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Iglewska-Nowak, I. Poisson Wavelets on \(n\)-Dimensional Spheres. J Fourier Anal Appl 21, 206–227 (2015). https://doi.org/10.1007/s00041-014-9366-x

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  • DOI: https://doi.org/10.1007/s00041-014-9366-x

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