Divergent Fourier series of integrable functions with respect to orthonormal systems

There exist two orthonormal systems such that the Fourier series of each functionƒ ∈ L[0, 1],ƒ ≠ 0, with respect to at least one of these systems diverges on a set of positive measure. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 717–724, May, 1998.     

Functional conditions for the convergence of Fourier series with respect to general orthonormal systems

In this paper we establish some functional conditions for the convergence of the Fourier series for functions from the class C(0, 1) with respect to general orthonormal systems.     

On a test of Jordan type for double Fourier series with respect to multiplicative systems

. , , .     

Generalized absolute convergence of single and double Fourier series with respect to multiplicative systems

Several sufficient conditions for generalized absolute convergence of single and double Vilenkin-Fourier series of bounded type are obtained. These conditions give multiplicative analogues of results due to L. Gogoladze and R. Meskhia, F. Móricz and A. Veres, M. Izumi and S. Izumi. Some of them are sharp in a certain sense.     

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)

On the order of magnitude of Fourier coefficients with respect to uniformly bounded orthonormal systems

пУсть {Φ _k (x) ∶k>0} — РАВНО МЕРНО ОгРАНИЧЕННАь ОРтОНОРМИРОВАННАь с ИстЕМА НА пРОстРАНстВЕX с кО НЕЧНОИ пОлОжИтЕльНО И МЕРОИ. Дль ФУНкцИИf ?L ^p (X) Дль НЕкОтОРОгО 1≤p≤2 ВВОДИМ ЕЕ кОЁФФИцИЕН ты ФУРьЕc _k (f) пО сИстЕМЕΦ _k (x). Дль жАДАННОгОα>0 М ы РАссМАтРИВАЕМ пОслЕДОВАтЕльНОсть {λ _k ^α :k ≥ 0} пОлОжИтЕльНых ЧИсЕл тАкИх, ЧтО т.Е. тАкИх, ЧтО гДЕA _n ^α — тАк НАжыВАЕМыЕ ЧИслА ЧЕжАРО. Мы ИжУЧАЕМ ЁтУ жАДАЧУ Дль ДВОИНых ОРтОгОНА льНых сИстЕМ, А тАкжЕ Дль НЕпРЕРыВНых ФУНкцИИ. пРИ ДОкАжАтЕльстВЕ М ы кОМБИНИРУЕМ МЕтОДы т ЕОРИИ сУММИРОВАНИь И ФУНкцИОНАльНОгО АН АлИжА.     

Classes of functions and Fourier coefficients with respect to complete orthonormal systems

This is the author review of a dissertation presented in the competition for the academic degree of Doctor of Physicomathematical Sciences. The dissertation was defended on October 10, 1974 in a meeting of the academic council of the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were: A. A. Talalyan, Corresponding Member of the Academy of Sciences of the Armenian SSR; N. P. Kuptsov, Doctor of Physicomathematical Sciences; and E. M. Nikishin, Doctor of Physicomathematical Sciences.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 445–463, September, 1976.     

The Fourier coefficients of continuous functions with respect to certain orthonormal systems

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened.     

Some classes of functions and Fourier coefficients with respect to general orthonormal systems

The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.