Presentations

Elements of wavelet analysis for students and postgraduates

Farkov Yu.A.

Russian Presidential Academy of National Economy and Public Administration, Department of Applied Information Technologies, Russia, 119571, Moscow, Vernadskogo Ave., 82. . E-mail: farkov-ya@ranepa.ru

Wavelet theory was developed in the 1980s and continues to evolve actively as part of modern harmonic analysis. Unlike the Fourier transform, continuous and discrete wavelet transforms are robust to noise effects and applicable for processing non-stationary signals. Frame theory developed in parallel with wavelet theory, and many frame constructions employ wavelets. Due to the completeness, stability, and redundancy of discrete signal representations, frames significantly complement orthogonal wavelet bases in areas such as signal analysis, image processing, data encoding and recovery, quantum information theory, and compressed sensing theory.

An introduction to wavelets and frames can be incorporated into university courses on mathematical analysis and linear algebra alongside elements of Fourier analysis. The report will discuss special courses developed using [1]–[5] and delivered by the author to students and postgraduates of the Faculty of Mechanics and Mathematics at Samara State University, the Faculty of Geophysics at the Russian State Geological Prospecting University, and the Faculty of Physics, Mathematics, and Natural Sciences at the Peoples' Friendship University of Russia. These special courses covered not only the fundamental concepts of wavelet analysis, the properties of classical wavelets, and tight frames in finite-dimensional spaces, but also included algorithms for discrete wavelet transforms defined using Walsh functions and their generalizations.

References

1. Farkov Yu.A., Introduction to Harmonic Analysis: From Fourier Series to Wavelets and Frames. – M.: MCCME, 2025. 112 p. (in Russian)

2. Farkov Yu.A., Tight Frames in Linear Algebra // Mathematics in Higher Education. 2020. No. 16, pp. 51-62. (in Russian)

3. Farkov Yu.A., Discrete Wavelet Transforms in Walsh Analysis // J. Math. Sci., New York. 2021. V. 257, No. 1, pp. 127–137.

4. Pereyra M.C., Ward L.A. Harmonic analysis. From Fourier to wavelets. Providence, RI: AMS; Princeton, NJ: IAS, 2012. 410 p. (Student Mathematical Library; V. 63).

5. Pinsky M.A. Introduction to Fourier analysis and wavelets. Providence, RI: AMS, 2002. 376 p. (Graduate Studies in Mathematics; V. 102).

• abstract in Russian (PDF)