For citation:
Bespalov M. S. Wavelet p-analogs of the discrete Haar transform. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 4, pp. 520-531. DOI: 10.18500/1816-9791-2021-21-4-520-531, EDN: EPNXLY
Wavelet p-analogs of the discrete Haar transform
Two $p$-analogs (for $p>2$) of the discrete version of the Haar system in vector symbolism are proposed and fast algorithms are constructed based on them. The main wavelet principles for constructing these Haar-like systems are proposed, such as the presence of several parent functions, $p$-ary dilations and shifts. One of the systems retains an orthogonality property. The calculation procedure has been simplified for another almost orthogonal system. The developed algorithms are presented with decimation in time, methods of their representation with decimation in frequency are indicated.
- Malozemov V. N., Macharskiy S. M. Osnovy discretnogo garmonicheskogo analiza [Basics of Discrete Harmonic Analysis]. St. Petersburg, Lan’, 2012. 304 p. (in Russian).
- Novikov I. J., Stechkin S. B. Basic wavelet theory. Russian Mathematical Surveys, 1998, vol. 53, iss. 6, pp. 1159–1231. https://dx.doi.org/10.1070/RM1998v053n06ABEH000089
- Novikov I. J., Protasov V. J., Skopina M. A. Teoriya vspleskov [Wavelet Theory]. Moscow, Fizmatlit, 2005. 616 p. (in Russian).
- Golubov B. I., Efimov A. V., Skvortsov V. A. Walsh Series and Transforms. Theory and Applications. Dordrecht, Boston, London, Kluver Academic Publishers, 1991. 360 p. (Russ. ed.: Moscow, Nauka, 1987. 344 p.).
- Bespalov M. S., Sklyarenko V. A. Diskretnye funktsii Uolcha i ikh prilogeniya [Discrete Walsh Functions and its Applications]. Vladimir, VlSU, 2014. 68 p. (in Russian).
- Zalmanzon L. A. Preobrazovanija Fur’e, Uolsha, Haara i ikh prilozhenija v upravlenii, svjazi i drugikh oblastjakh [Fourier, Walsh, Haar Transforms and Their Applications in Control, Communication and Other Fields]. Moscow, Nauka, 1989. 496 p. (in Russian).
- Dagman Je. E., Kucharev G. A. Bystrye diskretnye ortogonalnye preobrazovaniya [Fast Discrete Orthogonal Transforms]. Novosibirsk, Nauka, 1983. 232 p. (in Russian).
- Masharsky S. M., Malozemov V. N. Haar spectra of discrete convolutions. Computational Mathematics and Mathematical Physics, 2000, vol. 40, no. 6, pp. 914–921.
- Bellman R. Introduction to Matrix Analysis. New York, Toronto, London, Mcgray-Hill Book Company, 1960. 372 p. (Russ. ed.: Moscow, Nauka, 1969. 368 p.). 10.
- Trahtman A. M., Trahtman V. A. Osnovy teorii diskretnykh signalov na konechnykh intervalakh [Foundations of the Theory of Discrete Signals at Finite Intervals]. Moscow, Sovetskoe radio, 1975. 208 p. (in Russian).
- Bespalov M. S. Bernoulli’s discrete periodic functions. Applied Discrete Mathematics, 2019, no. 43, pp. 16–36. https://doi.org/10.17223/20710410/43/2
- Lukomskii S. F. Haar series on compact zero-dimensional abelian group. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2009, vol. 9, iss. 1, pp. 14–19 (in Russian). https://doi.org/10.18500/1816-9791-2009-9-1-14-19
- Bespalov M. S. Ternary discrete wavelet basis. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 3, pp. 367–377 (in Russian). https://doi.org/10.18500/1816-9791-2020-20-3-367-377
- Bespalov M. S. Discrete Chrestenson transform. Problems of Information Transmission, 2010, vol. 46, no. 4, pp. 353–375. https://doi.org/10.1134/S003294601004006X
- Efimov A. V. A bound on Fourier coefficients according to the Chrestenson – Levy system in ag-locally integral metric. Mathematical Notes, 1990, vol. 48, iss. 4, pp. 992–997. https://doi.org/10.1007/BF01139598
- 1102 reads