Abstract
For the discrete Fourier transform with respect to the system of characters of a local field with zero characteristic, we propose a fast algorithm. We find the complexity of the algorithm.
Similar content being viewed by others
References
H. Jiang, D. Li and N. Jin, “Multiresolution analysis on local fields,” J. Math. Anal. Appl. 294 (2), 523–532 (2004).
B. Behera and Q. Jahan, “Multiresolution analysis on local fields and characterization of scaling functions,” Adv. Pure Appl. Math. 3 (2), 181–202 (2012).
B. Behera and Q. Jahan, “Biorthogonal wavelets on local fields of positive characteristic,” Commun. Math. Anal. 15 (2), 52–75 (2013).
B. Behera and Q. Jahan, “Wavelet packets and wavelet frame packets on local fields of positive characteristic,” J. Math. Anal. Appl. 395 (1), 1–14 (2012).
B. Behera and Q. Jahan, “Characterization of wavelets and MRA wavelets on local fields of positive characteristic,” Collect. Math. 66 (1), 33–53 (2015).
S. F. Lukomskii and A. M. Vodolazov, “Non-Haar MRA on local fields of positive characteristic,” J. Math. Anal. Appl. 433 (2), 1415–1440 (2016).
G. S. Berdnikov, I. S. Kruss and S. F. Lukomskii, “On orthogonal systems of shifts of scaling function on local fields of positive characteristic,” Turkish J. Math. 41 (2), 244–253 (2017).
F. A. Shah, S. Sharma and M. Y. Bhat, “Wavelet frame characterization of Lebesgue spaces on local fields,” An. Stiint. Univ. Al.I. Cuza Iasi. Mat. (N.S.), T. LXIV, f. 2, 429–445 (2018).
S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izvestiya Math. 66 (2), 367–376 (2002).
S. Albeverio, S. Evdokimov, M. Skopina, “p-Adic multiresolution analysis and wavelet frames,” J. Fourier Anal. Appl. 16 (5), 693–714 (2010).
S. Evdokimov and M. Skopina, “On orthogonal p-adic wavelet bases,” J. Math. Anal. Appl. 424 (2), 952–965 (2015).
S. Evdokimov, “On non-compactly supported p-adic wavelets,” J. Math. Anal. Appl. 443 (2), 1260–1266 (2016).
R. L. Benedetto, “Examples of wavelets for local fields,” in Wavelets, Frames and Operator Theory, Contemp. Math. 345, 27–47 (Am. Math. Soc., Providence, 2004)
V. M. Shelkovich and M. Skopina, “p-Adic Haar multiresolution analysis and pseudo-differential operators,” J. Fourier Anal. Appl. 15 (3), 366–393 (2009).
A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and their application to pseudodifferential operators and equations,” Appl. Comput. Harmon. Anal. 28 (1), 1–23 (2009).
S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich, Theory of p-Adic Distributions: Linear and Nonlinear Models (London Math. Society, Cambridge Univ. Press, 2010).
S. F. Lukomskii and A. M. Vodolazov, “Fast discrete Fourier transform on local fields of positive characteristic,” Probl. Inform. Transm. 53 (2), 155–163 (2017).
J. Cassel and A. Frohlich, Eds., Algebraic Number Theory (Academic Press, London and New York, 1967).
M. S. Bespalov, “Discrete Chrestenson transforms,” Probl. Inform. Transm. 46 (4), 353–375 (2010).
M. H. Taibleson, Fourier Analysis on Local Fields (Prinston Univ. Press and Univ. of Tokyo Press, Prinston, New Jersey, 1975).
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions (Spriger-Verlag, New York, Heidelberg, Berlin, 1977).
B. Golubov, A. Efimov and V. Skvortsov, Walsh Series and Transforms. Theory and Applications (Springer-Science+Business Media, B. V., 1991).
Author information
Authors and Affiliations
Corresponding authors
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Lukomskii, S.F., Vodolazov, A.M. Fast Discrete Fourier Transform on Local Fields of Zero Characteristic. P-Adic Num Ultrametr Anal Appl 12, 39–48 (2020). https://doi.org/10.1134/S2070046620010045
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046620010045