Localization for multiple Fourier series of functions of bounded harmonic variation

An example of a function of four variables belonging to the class of bounded harmonic variation (in a weak sense) is constructed. Cubic sums of a trigonometric Fourier series for this function does not possess the localization property.     

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ _k и их разностейΔλ _k=λ _k?λ _k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS _m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.     

Interconnection of Cesaro methods and discrete Riesz means

Inclusion problems of Cesaro methods and discrete Riesz means are discussed.     

On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series

- - , .     

Tauberian conditions connecting Cesaro and Riesz discrete means methods

Tauberian conditions for Cesaro methods and discrete Riesz means are discussed.     

On the divergence of linear summation methods of Fourier series

[0, 1],fL(0,2),

Approximation of classes of convolutions by linear methods of summation of Fourier series

We consider a family of special linear methods of summation of Fourier series and establish exact equalities for the approximation of classes of convolutions with even and odd kernels by polynomials generated by these methods.     

Summability of multiple trigonometric Fourier series by linear methods

The paper deals with the problem of estimation of deviations of functions of several variables from linear means of their multiple trigonometric Fourier series. An approach of reducing this problem to the corresponding problem for functions of single variable is developed.     

On absolute summation of Fourier series by subsequences

В работе изучается сл едующая задача. Пусть заданы числа 0<α≦1 и β<α. При каки х условиях на строго во зрастающую последов ательность натуральных чисел {n _k } _k ^t8 =1 для всех 2π-периодических функ ций \(f(x) \sim \sum\limits_{v = - \infty }^\infty {c_v e^{ivx} } \) , принадлежащих к лассу Lip α, равномерно пох будет выполнено неравенство $$\sum\limits_{k = 1}^\infty {|\sum\limits_{n_k \leqq |v|< n_{k + 1} } {c_v e^{ivx} } |n_k^\beta< \infty ?} $$ .     

Asymptotic formulas for the deviation of linear summation methods of double Fourier series

The article investigates a wide class of trigonometric polynomial approximations to periodic functions of two variables which includes, in particular, Fejer and Riesz averages. Asymptotic formulas for the deviation of these methods in the uniform metric are derived.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 189–196, 1978.In conclusion, I would like to acknowledge the valuable attention of V. V. Zhuk.     

Summation of arbitrary series by Riesz methods

It is known (theorem of Agnew and Darevskii) that for each divergent real sequence {s} and each real number c, there exists a T-method of summing {s_n} to c. In this note it is shown that for each divergent sequence which is bounded above or below we can take the T-method in the above theorem to be a Riesz method. We also study Riesz summability of unbounded (above and below) sequences.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 541–550, November, 1968.