This review paper is intended to give a useful guide for those who want to apply the discrete wavelet transform in practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to the corresponding literature. The multiresolution analysis and fast wavelet transform have become a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for the achievement of a goal. Analysis of various functions with the help of wavelets allows one to reveal fractal structures, singularities etc. The wavelet transform of operator expressions helps solve some equations. In practical applications one often deals with the discretized functions, and the problem of stability of the wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves to a few examples only. The authors would be grateful for any comments which would move us closer to the goal proclaimed in the first phrase of the abstract.

PACS: 02.60.−x, 02.70.−c, 05.40.−a, 87.57.−s (all) DOI: 10.1070/PU2001v044n05ABEH000918 URL: https://ufn.ru/en/articles/2001/5/a/ 000170938900001 Citation: Dremin I M, Ivanov O V, Nechitailo V A "Wavelets and their uses" Phys. Usp. 44 447–478 (2001) BibTexBibNote ® (generic) BibNote ® (RIS)MedlineRefWorks TY JOUR TI Wavelets and their uses AU Dremin, I. M. AU Ivanov, O. V. AU Nechitailo, V. A. PB Physics-Uspekhi PY 2001 JO Physics-Uspekhi JF Physics-Uspekhi JA Phys. Usp. VL 44 IS 5 SP 447-478 UR https://ufn.ru/en/articles/2001/5/a/ ER https://doi.org/10.1070/PU2001v044n05ABEH000918 Оригинал: Дремин И М, Иванов О В, Нечитайло В А «Вейвлеты и их использование» УФН 171 465–501 (2001); DOI: 10.3367/UFNr.0171.200105a.0465