On the uniform convergence of trigonometric series

In this paper, we obtain sufficient conditions for the uniform convergence of trigonometric series with monotone (in the extended sense) coefficients.     

On the convergence in mean of trigonometric Fourier series

We prove the sharpness of Zygmund’s theorem, which asserts that if a 2π-periodic function f belongs to L ln⁺ L, then its Fourier series is convergent in mean.     

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

Accelerating the convergence of trigonometric series

A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.     

On the convergence of greedy approximants of trigonometric Fourier series

Efficient versions of the Cauchy criterion for the convergence in L _p of greedy approximants of trigonometric Fourier series are discussed.     

On convergence of multiple trigonometric series with monotone coefficients

We study the Pringsheim pointwise convergence of multiple trigonometric series. We obtain a condition on the coefficients of the series that guarantees its Pringsheim convergence and prove the unimprovability of this condition.     

On sets of absolute convergence for multiple trigonometric series

We obtain a sufficient condition for a set of measure zero in N-dimensional space to be a set of absolute convergence (A.C. set) for an N-tuple trigonometric series. We also show that, in a certain subclass of sets of measure zero (namely in the subclass of monotonic curves), this condition cannot be sharpened.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 625–635, May, 1973.The author wishes to thank E. M. Nikishin for some valuable remarks.     

The best convergence of multiple trigonometric series

We consider the best convergence of multiple trigonometric series. We indicate essential distinction of the behavior (in this sense) of multiple series from that of simple ones. In particular, the well-known result obtained by S. N. Bernshtein on the best convergence of a series with an odd ratio of frequencies does not hold for a multiple series in the case of the approximation by polynomials with harmonics from rectangles (in the sense of Pringsheim), but it is true for “angular” approximations.     

On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

For a trigonometric series

Sets of absolute convergence of double trigonometric series

We obtain a sufficient condition for a set of plane measure zero to be a set of absolute convergence (an A.C.-set) for a double trigonometric series. Specifically, let y=f(x) (0 x2) be a smooth curve and let inif (x) t <. Then, any set of positive linear measure lying on this curve is an A.C.-set.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 473–480, May, 1972.In conclusion, I am indebted to E. M. Nikishin for suggesting the problem.     

On necessary conditions for the convergence of Fourier series

We establish necessary conditions for the convergence of multiple Fourier series of integrable functions in the mean.     

On the best convergence of trigonometric integrals

ОсНОВНОИ РЕжУльтАт Ё тОИ стАтьИ жАклУЧАЕт сь В слЕДУУЩЕМ. ЕслИ с ФИксИРОВАННыМr=2, 3,..., тО Дль тОгО, ЧтОБы ИМЕлО М ЕстО (*) гДЕ ДОстАтОЧНО ВыпОлНЕН ИЕ слЕДУУЩИх УслОВИИ: 1) ЕслИ $$1< \frac{{n_\nu + 1}}{{n_\nu }} = g_\nu \in N, \nu = 0,1,$$ тО Дль кАжДОгОΝ=0,1,... ИМЕ Ет МЕстО (**) 2) ЕслИ тО (***) $$n_{\nu + 1} \geqslant (2r - 1)n_\nu , \nu = 0,1,...$$ . ЕслИ И тО УслОВИь (**), (***) ьВльУтсь тАкжЕ НЕОБхОДИМыМИ Д ль (*).     

On the boundedness below of trigonometric polynomials of best approximation

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α _n } _n=0 ^∞ , are uniformly bounded below if the conditions a _n = O(n ^?1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent ?1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L _2π ¹ .     

Integrability conditions for multiple trigonometric series

Sufficient conditions (the Boas-Telyakovskii conditions) on the coefficients of multiple trigonometric series are found that guarantee integrability of the sums of these series. Under these conditions, estimates are obtained for integrals of the moduli of the functions defined by multiple trigonometric series. In addition, the Boas-Telyakovskii conditions are compared with previously known conditions. It is shown that the Boas-Telyakovskii conditions are the most general ones.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 340–365, March, 1992.     

On the convergence of generalized Cesàro means of trigonometric Fourier series. I

The behavior of generalized Cesàro (C, α _n )-means (α _n ∈ (0, d), d > 0) of trigonometric Fourier series of H ^ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.     

On the convergence of generalized Cesàro means of trigonometric Fourier series. II

The behavior of generalized Cesàro (C, α _n )-means (α _n ∈ (?1,0)) of trigonometric Fourier series of H ^ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.