On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

On conjugate functions and trigonometric series

The present paper is an abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical and Mathematical Sciences. The dissertation was defended on June 30, 1967 before the Academic Council of the Faculty of Mechanics and Mathematics of the M. V. Lomonosov Moscow State University. The official opponents were Professor A. V. Efimov, Doctor of Physical and Mathematical Sciences, Professor V. A. Il'in, Doctor of Physical and Mathematical Sciences, and Professor G. P. Tolstov, Doctor of Physical and Mathematical Sciences.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 695–702, December, 1967.     

On some properties of trigonometric series with monotone decreasing coefficients

В статье рассматрива ются одномерные и дву мерные тригонометрические ряды с моно-тонными коэффициентами. Дает ся пример двойного тригонометрическог о ряда (1) $$\mathop \sum \limits_{n,k = 1}^\infty a_{nk} \sin nx\sin ky,$$ , коэффициенты которо го монотонны поk и поп, любая последовательность \(\{ S_{n_k m_k } (x,y)\} _{k = 1}^\infty\) прямоугольных части чных сумм ряда (1), где min(n _k ,m _k )→∞ приk→∞, расходится по чти всюду на (0,n)². Кроме того, изучается мера множеств нулей ф ункций (2) $$f(x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{n = 1}^{a_0 } a_n \cos nx\tilde f(x) = \mathop \sum \limits_{n = 1}^\infty a_n \sin nx,$$ , гдеа _n ↓ приn→ ∞, и доказ ьшается несколько те орем о скорости убывания ко эффициентовa _n рядов (2), если все част ичные суммыS _n (f,x) или \(S_n (\tilde f,x)\) дляn=1,2,... неотрицате ль-ны на (0,n).     

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

On uniformly convergent rearrangements of trigonometric Fourier series

We show that if the module of continuity ω(ƒ, δ) of a 2π-periodic function ƒ ∈ {ie081-01} is o(1/ log log 1/δ) as δ → 0+, then there exists a rearrangement of the trigonometric Fourier series of ƒ converging uniformly to ƒ. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

On some local properties of the sum of lacunary trigonometric series

В статье изучается по ведение суммы лакуна рного тригонометрическог о ряда при приближени и к некоторой фиксиров анной произвольной т очке. Первая половина рабо ты посвящена изложен ию метода исследования локаль ных свойств суммы лакунарного ря да, разработанного ав тором. Вторая половина рабо ты посвящена приложе ниям этого метода. Здесь в частно сти, получаются необходи мые и достаточные усл овия для интегрируемости сум мы лакунарного ряда с весом при широк их условиях на вес. При ведем соответствующий рез ультат. Пусть?р(x) — сумма ряда \(a + \sum\limits_{n = 1}^\infty {a_n \cos (\lambda _n x + \psi _n )} \) , гдеа, а _n ,λ _n ,ψ _n — действительные числа,εa _n ^/2 <∞,a _n ≧0,λ _n >0 приn≧1 и \(\mathop {\inf }\limits_{n \geqq 1} \lambda _{n + 1} /\lambda _n > 1\) . При этих условиях функция?(х) определена почти всю ду. Пустьр>0 иω(х) — положительная неуб ывающая функция, определенная при все хх>0, которая при некот оромC>0 удовлетворяет услов ию:ω(2x)≦ ≦Cω(х) при всехх>0. Тогда имеет место Теорема. Для того, чтоб ы интеграл \(\int\limits_{ + 0} {|\varphi (x)|^p \frac{{dx}}{{\omega (x)}}} \) сходился, необходимо и достато чно, чтобы сходились все р яды $$\begin{gathered} \sum\limits_{n = 1}^\infty {D_n (\sum\limits_{k = n}^\infty {a_k^2 } )^{p/2} ,} \sum\limits_{n = 2}^\infty {D_n |a_n + \sum\limits_{k = 1}^{n - 1} {a_k \cos } \psi _k |^p ,} \hfill \\ \sum\limits_{n = 2}^\infty {D_n (pj)|\sum\limits_{k = 1}^{n - 1} {a_k \lambda _k^j \cos (\psi _k + \pi j/2)} |^p ,} j = 1,2,..., \hfill \\ \end{gathered} $$ , где $$D_n = \int\limits_{I_n } {\frac{{dx}}{{\omega (x)}},} D_n (pj) = \int\limits_{I_n } {\frac{{x^{pj} dx}}{{\omega (x)}},} a I_n = [\pi \lambda _n^{ - 1} ,\pi \lambda _{n - 1}^{ - 1} ]$$      

The question of A*-summability of double trigonometric Fourier series

It is proved that for anyf(x, y) L(R), where R=[-,,-, ], a function (x, y), exists such that ¦(x, y) ¦=¦f(x, y) ¦ for almost all (x, y) R. The Fourier series of the function (x, y) and all conjugate trigonometric series are A^*-summable almost everywhere.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 145–150, February, 1972.     

On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.     

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 Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 275–279, February, 1992.     

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 uniformly convergent double Fourier series

The uniform convergence of the partial sums, relative to a system of homothetic starlike polygons, is considered. Among other things, it is established an arbitrary preassigned positive sequence from can be majorized by the coefficients of a uniformly convergent double Fourier series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 151–162, 1989.     

Study of some trigonometric series

In this paper we generalize a result due to Hardy. We present a simple example of an everywhere divergent trigonometric series over the squares of the positive integers with coefficients which tend to zero.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 481–492, April, 1973.     

Weighted integrability of double trigonometric series

We study the double trigonometric series whose coefficients are such that Then its rectangular partial sums converge uniformly to some . We give sufficient conditions for the Lebesgue integrability of , where , or . For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.

     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).