Advertisement

Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 2923–2956 | Cite as

Gibbs Phenomenon of Framelet Expansions and Quasi-projection Approximation

Article

Abstract

The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the nth Fourier partial sums overshoot a target function at jump discontinuities in such a way that such overshoots do not die out as n goes to infinity. The Gibbs phenomenon for wavelet expansions using (bi)orthogonal wavelets has been studied in the literature. Framelets (also called wavelet frames) generalize (bi)orthogonal wavelets. Approximation by quasi-projection operators are intrinsically linked to approximation by truncated wavelet and framelet expansions. In this paper we shall establish a key identity for quasi-projection operators and then we use it to study the Gibbs phenomenon of framelet expansions and approximation by general quasi-projection operators. We shall also study and characterize the Gibbs phenomenon at an arbitrary point for approximation by quasi-projection operators. As a consequence, we show that the Gibbs phenomenon appears at all points for every tight or dual framelet having at least two vanishing moments and for quasi-projection operators having at least three accuracy orders. Our results not only improve current results in the literature on the Gibbs phenomenon for (bi)orthogonal wavelet expansions but also are new for framelet expansions and approximation by quasi-projection operators.

Keywords

Gibbs phenomenon Quasi-projection operators Dual multiframelets (Bi)orthogonal multiwavelets Vanishing moments Approximation order Accuracy order Polynomial reproduction 

Mathematics Subject Classification

42C40 42C15 41A25 41A35 65T60 

Notes

Acknowledgements

The author would like to thank the reviewers for their valuable suggestions which improved the presentation of the paper.

References

  1. 1.
    Chui, C.K., He, W., Stöckler, J.: Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13, 224–262 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Series in Applied Mathematics, vol. 61. SIAM (1992)Google Scholar
  3. 3.
    Daubechies, I., Han, B.: Pairs of dual wavelet frames from any two refinable functions. Constr. Approx. 20, 325–352 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gibbs, J.W.: Letter to the editor. Nature 59, 606 (1899)CrossRefGoogle Scholar
  6. 6.
    Gomes, S.M., Cortina, E.: Some results on the convergence of sampling series based on convolution integrals. SIAM J. Math. Anal. 26, 1386–1402 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gottlieb, D., Shu, C.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–688 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Han, B.: On dual wavelet tight frames. Appl. Comput. Harmon. Anal. 4, 380–413 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Han, B.: Dual multiwavelet frames with high balancing order and compact fast frame transform. Appl. Comput. Harmon. Anal. 26, 14–42 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Han, B.: Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space. Appl. Comput. Harmon. Anal. 29, 330–353 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Han, B.: Nonhomogeneous wavelet systems in high dimensions. Appl. Comput. Harmon. Anal. 32, 169–196 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Han, B.: Homogeneous wavelets and framelets with the refinable structure. Sci. China Math. 60, 2173–2198 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Han, B.: Framelets and Wavelets: Algorithms, Analysis, and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham (2017)CrossRefGoogle Scholar
  14. 14.
    Han, B., Mo, Q.: Multiwavelet frames from refinable function vectors. Adv. Comput. Math. 18, 211–245 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jerri, A.: The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  16. 16.
    Jetter, K., Zhou, D.-X.: Order of linear approximation from shift-invariant spaces. Constr. Approx. 11, 423–438 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jia, R.-Q.: Approximation with scaled shift-invariant spaces by means of quasi-projection operators. J. Approx. Theory 131, 30–46 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jia, R.-Q., Jiang, Q.-T.: Approximation power of refinable vectors of functions. Wavelet analysis and applications. In: AMS/IP Studies in Advanced Mathematics, 25, pp. 155–178. American Mathematical Society, Providence, RI (2002)Google Scholar
  19. 19.
    Kelly, S.E.: Gibbs phenomenon for wavelets. Appl. Comput. Harmon. Anal. 3, 72–81 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mohammad, M., Lin, E.-B.: Gibbs phenomenon in tight framelet expansions. Commun. Nonlinear Sci. Numer. Simul. 55, 84–92 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ron, A., Shen, Z.: Affine systems in \(L_2({\mathbb{P}}^d)\): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ruch, D.K., Van Fleet, P.J.: Gibbs’ phenomenon for nonnegative compactly supported scaling vectors. J. Math. Anal. Appl. 304, 370–382 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shen, X.: Gibbs phenomenon in orthogonal wavelet expansion. J. Math. Study 35(4), 343–357 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Shen, X.: Gibbs phenomenon for orthogonal wavelets with compact support. In: Jerri, J. (ed.) Advances in the Gibbs Phenomenon, pp. 337–369. Sampling Publishing, Potsdam (2011)Google Scholar
  25. 25.
    Shim, H.-T., Volkmer, H.: On the Gibbs phenomenon for wavelet expansions. J. Approx. Theory 84, 74–95 (1996)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wilbraham, H.: On a certain periodic function. Camb. Dublin Math. J. 3, 198–201 (1848)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations