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Explicit formula for MRA-wavelets on local fields

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Abstract

We provide an explicit formula for the wavelets associated with a multiresolution analysis of \(L^2(K)\), where K is a local field of positive characteristic. We also give several examples to illustrate this result.

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References

  1. Albeverio, S., Evdokimov, S., Skopina, M.: \(p\)-adic multiresolution analysis and wavelet frames. J. Fourier Anal. Appl. 16, 693–714 (2010)

    Article  MathSciNet  Google Scholar 

  2. Behera, B.: Haar wavelets on the Lebesgue spaces of local fields of positive characteristic. Colloq. Math. 136(2), 149–168 (2014)

    Article  MathSciNet  Google Scholar 

  3. Behera, B.: Wavelet sets and scaling sets in local fields (preprint)

  4. Behera, B., Jahan, Q.: Wavelet packets and wavelet frame packets on local fields of positive characteristic. J. Math. Anal. Appl. 395, 1–14 (2012)

    Article  MathSciNet  Google Scholar 

  5. Behera, B., Jahan, Q.: Multiresolution analysis on local fields and characterization of scaling functions. Adv. Pure Appl. Math. 3, 181–202 (2012)

    Article  MathSciNet  Google Scholar 

  6. Behera, B., Jahan, Q.: Characterization of wavelets and MRA wavelets on local fields of positive characteristic. Collect. Math. 66, 33–53 (2015)

    Article  MathSciNet  Google Scholar 

  7. Ciarlet, P.G.: Introduction to Numerical Linear Algebra and Optimisation. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  8. Jia, R.Q., Shen, Z.: Multiresolution and wavelets. Proc. Edinb. Math. Soc. 37, 271–300 (1994)

    Article  MathSciNet  Google Scholar 

  9. Jiang, H., Li, D., Jin, N.: Multiresolution analysis on local fields. J. Math. Anal. Appl. 294, 523–532 (2004)

    Article  MathSciNet  Google Scholar 

  10. Khrennikov, AYu., Shelkovich, V.M., Skopina, M.: \(p\)-adic refinable functions and MRA-based wavelets. J. Approx. Theory 161, 226–238 (2009)

    Article  MathSciNet  Google Scholar 

  11. Lukomskii, S.F.: Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases. Sb. Math. 201, 669–691 (2010)

    Article  MathSciNet  Google Scholar 

  12. Lukomskii, S.F.: Multiresolution analysis on product of zero-dimensional abelian groups. J. Math. Anal. Appl. 385, 1162–1178 (2012)

    Article  MathSciNet  Google Scholar 

  13. Ramakrishnan, D., Valenza, R.: Fourier Analysis on Number Fields. Springer, New York (1999)

    Book  Google Scholar 

  14. Taibleson, M.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  15. Weil, A.: Basic Number Theory. Springer, New York (1974)

    Book  Google Scholar 

  16. Zalik, R.A.: Bases of translates and multiresolution analyses. Appl. Comput. Harmon. Anal. 24, 41–57 (2008) [Corrigendum: Appl. Comput. Harmon. Anal. 29, 121 (2010)]

  17. Zalik, R.A.: On orthonormal wavelet bases. J. Comput. Anal. Appl. 27, 790–797 (2019)

    Google Scholar 

  18. Zalik, R.A.: On multiresolution analyses of multiplicity \(n\). J. Comput. Anal. Appl. 29, 1055–1062 (2021)

    Google Scholar 

  19. Zheng, W.X., Su, W.Y., Jiang, H.K.: A note to the concept of derivatives on local fields. Approx. Theory Appl. 6, 48–58 (1990)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Biswaranjan Behera.

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Communicated by Ferenc Weisz.

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Behera, B. Explicit formula for MRA-wavelets on local fields. Adv. Oper. Theory 6, 55 (2021). https://doi.org/10.1007/s43036-021-00152-3

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  • DOI: https://doi.org/10.1007/s43036-021-00152-3

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