The problems which were studied in this project are central to Mathematical Analysis. They involve  techniques  which  have  been  developed  a  great  deal  during the last three decades.  Unlike the classical theory of Fourier series which deals with decomposition of a function into continuous waves, the Vilenkin (Walsh) functions are “rectangular waves”. Such waves have already been used frequently in the theory of signal transmission, filtering, image enhancement and digital signal processing.

According to the proposal aims and problems, widely was used methods of real analysis combined with methods of abstract and non-linear harmonic analysis together with theory of approximation. Other research methods include theory of function spaces. By applying these methods we have proved boundedness of some operators in the martingale Hardy spaces.

In the frame of this project we have published 6 scientific publications in international peer-reviewed (Impact Factor) journals. These results certainly draw the attention of researchers interested in dyadic harmonic analysis. Another significant value of the realized project is that it gave possibility to intensify my professional international contacts and 4 publications were written by collaboration with foreign mathematicians, which is very important for every young students on the beginning step of the research.

List of publications:

1. G. Tephnadze, On the partial sums of Walsh-Fourier series, Colloq. Math., 141, 2 (2015), 227-242.
2. N. Memić, I. Simon, G. Tephnadze, Strong convergence of two-dimensional Vilenkin-Fourier series, Math. Nachrichten, 289, 4 (2016) 485–500.
3. K. Nagy, G. Tephnadze, Strong convergence theorem for Walsh-Marcinkiewicz means, Math. Inequal. Appl., 19, 1 (2016), 185–195.
4. L.E. Persson, G. Tephnadze, A sharp boundedness result concerning some maximal operators of Vilenkin-Fejér means, Mediterr. J. Math., DOI: 10.1007/s00009-015-0565-8.
5. I. Blahota, L.E. Persson, G. Tephnadze, On the Lebesgue constants with respect to Vilenkin systems two-sided estimates of the Lebesgue constants with respect to Vilenkin systems and applications, Glasgow Math. J., DOI: 10.1017/S0017089516000549.
6. G. Tephnadze, On the partial sums with respect to Vilenkin-Fourier systems, J. Contemp. Math. Anal., (to appear).