Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Bespalov M. S. Ternary Discrete Wavelet Basis. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 3, pp. 367-377. DOI: 10.18500/1816-9791-2020-20-3-367-377, EDN: TTKRGK

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2020
Full text:
download
(downloads: 495)
Language: 
Russian
Heading: 
Computer Sciences
Article type: 
Article
UDC: 
519.72
DOI: 
10.18500/1816-9791-2020-20-3-367-377
EDN: 
TTKRGK

Ternary Discrete Wavelet Basis

Autors: 
Bespalov Mikhail S., Vladimir State University
Abstract: 

The discrete version and the basic construction of the ternary multiresolution analysis are given, similar to the binary model case of the Haar multiresolution analysis. Based on the constructed basis, an algorithm similar to the fast Haar transformation is proposed. Typical calculation examples are provided.

Key words: 
discrete harmonic analysis
discrete Haar transform
wavelet
shifts and dilations
multiresolution analysis
References: 
  1. Novikov I. Ya., Protasov V. Yu., Skopina M. A. Teoriya vspleskov [Wavelet theory]. Moscow, Fizmatlit, 2005. 616 p. (in Russian).
  2. Masharsky S. M., Malozemov V. N. Haar spectra of discrete convolutions. Comput. Math. Math. Phys., 2000, vol. 40, no. 6, pp. 914–921.
  3. Ber M. G., Malozemov V. N. The best formulae for the approximate computation of discrete Fourier transforms. Comput. Math. Math. Phys., 1992, vol. 32, no. 11, pp. 1533– 1544.
  4. Bespalov M. S. Bernoulli’s discrete periodic functions. Applied Discrete Mathematics, 2019, no. 43, pp. 16–36. DOI: https://doi.org/10.17223/20710410/43/2
  5. Malozemov V. N., Macharskiy S. M. Osnovy discretnogo garmonicheskogo analiza [Basics of Discrete Harmonic Analysis]. St. Petersburg, Lane, 2012. 304 p. (in Russian).
  6. Bespalov M. S., Sklyarenko V. A. Diskretnye funktsii Uolsha i ikh prilogeniya [Discrete Walsh Functions and its Applications]. Vladimir, Vladimirskiy gosudarstvennyi universitet, 2014. 68 p. (in Russian).
  7. Farkov Yu. A. Orthogonal Wavelets on Locally Compact Abelian Groups. Funct. Anal. Appl., 1997, vol. 31, iss. 4, pp. 294–296. DOI: https://doi.org/10.1007/BF02466067
  8. Lang W. C. Wavelet analysis on the Cantor dyadic group. Housten J. Math., 1998, vol. 24, no. 3, pp. 533–544.
  9. Lukomskii S. F. Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases. Sb. Math., 2010, vol. 201, no. 5, pp. 669–691. DOI: http://dx.doi.org/10.1070/SM2010v201n05ABEH004088
  10. Pleshcheva E. A., Chernykh N. I. Construction of orthogonal multiwavelet bases. Proc. Steklov Inst. Math. (Suppl.), 2015, vol. 288, suppl. 1, pp. 162–172. DOI: https://doi.org/10.1134/S0081543815020169
  11. Farkov Yu. A., Stroganov S. A. The use of discrete dyadic wavelets in image processing. Russian Math. (Iz. VUZ), 2011, vol. 55, no. 7, pp. 47–55. DOI: https://doi.org/10.3103/S1066369X11070073
  12. Barichev A. A., Lukomskii D. S., Lukomskii S. F. Systems of Scales and Shifts in the Problem Still Image Compression. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 4, pt. 2, pp. 505–510 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2014-14-4-505-510
  13. Daubechies I. Ten Lestures on Wavelets. SIAM, 1992. 454 p. CBMS-NSR Series in Appl. Math.
  14. Bespalov M. S. Discrete Chrestenson transform. Probl. Inform. Transm., 2010, vol. 46, iss. 4, pp. 353–375. DOI: https://doi.org/10.1134/S003294601004006X
  15. Kolmogorov A. N., Fomin S. V. Elementy teorii funktsiy i funktsional’nogo analiza [Elements of the theory of functions and functional analysis]. Moscow, Nauka, 1976. 544 p. (in Russian).
Received: 
06.05.2019
Accepted: 
31.12.2019
Published: 
31.08.2020
  • 1511 reads