Restricted maximal operators of Fejér means of double Walsh-Fourier series

The main aim of this paper is to prove that there exists a martingale f ∈ H ₁^2/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier series does not belong to the space weak-L _1/2.     

Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series

For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.     

Triangular Fejér summability of two-dimensional Walsh-Fourier series

It is proved that the operators σ _n ^Δ of the triangular-Fejér-means of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space H _p to L _p for all 4/5 < p≤∞.     

Pointwise convergence of Marcinkiewicz-Fejér means of double Vilenkin-Fourier series

In this paper we give a characterization of points at which the Marcinkiewicz-Fejér means of double Vilenkin-Fourier series converge.     

Extension of a theorem of Fejér to double Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π^-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:

The martingale hardy type inequality for Marcinkiewicz-Fejér means of two-dimensional conjugate Walsh-Fourier series

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H _p such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L _p .     

Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series

In this article we discuss the Nörlund means of cubical partial sums of Walsh-Fourier series of a function in L ^p (1 ≤ p ≤ ∞). We investigate the rate of the approximation by this means, in particular, in Lip(α, p), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞ by L ^p we mean C _W , the collection of the uniformly W-continuous functions. Our main theorems state that the approximation behavior of the two-dimensional Walsh- Nörlund means is so good as the approximation behavior of the one-dimensional Walsh- Nörlund means. As special cases, we get the Nörlund logarithmic means of cubical partial sums of Walsh-Fourier series discussed recently by Gát and Goginava [5] in 2004 and the (C, β)-means of Marcinkiewicz type with respect to double Walsh-Fourier series discussed by Goginava [10]. Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series was given by Móricz and Siddiqi [14].     

Approximation of functions of Dirichlet class by Fejér means

For the Dirichlet classes D _p of holomorphic functions in the disk, we obtain the exact orders of best polynomial approximations and of upper bounds for deviations of Fejér means of Taylor series in the Hardy spaces H _p.     

On the maximal operators of Walsh-Kaczmarz-Fejér means

The main aim of this paper is to prove that the maximal operator $\sigma _p^{\kappa , * } f: = \sup _{n \in P} {{\left| {\sigma _n^\kappa f} \right|} \mathord{\left/ {\vphantom {{\left| {\sigma _n^\kappa f} \right|} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-0em} {p - 2}}} }}} \right. \kern-0em} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-0em} {p - 2}}} }}$ is bounded from the Hardy space H _p to the space L _p for 0 < p < 1/2.     

Fejér type theorems for Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2], then the nth partial sum of its formally differentiated Fourier series divided by n converges to ^-1 [F(x+0) - F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of nonperiodic functions of bounded variation is also known. These theorems can be interpreted in such a way that the terms of the Fourier-Stieltjes (or Fourier) series of F determine the atoms of the finite Borel measure on the torus T:= [0, 2) induced by an appropriate extension of F (or by F itself in the periodic case). The aim of the present paper is to extend all of these results to the Cesàro as well as Abel-Poisson means of Fourier-Stieltjes (or Fourier) series of a nonperiodic (or periodic) function F of bounded variation. At the end, we sketch a possible extension of these results to linear means defined by more general kernels.     

Best constants for approximations of periodic functions by Fejér operators

We obtain the best estimate for approximations of continuous 2-periodic functions by Fejér summation operators. The estimate is expressed in terms of the moduli of continuity of functions to be approximated. In the class of functions satisfying Lipschitz condition, we obtain the best constant for the approximation by Fejer integral and summation operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 75–82, 1990.     

Almost Everywhere Strong Summability of Fejér Means of Rectangular Partial Sums of Two-dimensional Walsh-Fourier Series

In this paper we prove a BMO-estimate for rectangular partial sums of two-dimensional Walsh-Fourier series, and using this result we establish almost everywhere exponential summability of rectangular partial sums of double Walsh-Fourier series.     

On the divergence of Nörlund logarithmic means of Walsh-Fourier series

It is well known in the literature that the logarithmic means 1/logn ^n-1∑k=1 Sk（f）/k of Walsh or trigonometric Fourier series converge a.e. to the function for each integrable function on the unit interval. This is not the case if we take the partial sums. In this paper we prove that the behavior of the so-called NSrlund logarithmic means 1/logn ^n-1∑k=1 Sk（f）/n-k is closer to the properties of partial sums in this point of view.     

Strong convergence theorems for Walsh–Fejér means

As main result we prove that Fejér means of Walsh–Fourier series are uniformly bounded operators from H _p to H _p (0<p≦1/2).     

Properties of Cesàro means of double Fourier series

The pointwise behavior of partial sums and Cesàro means of trigonometric series were studied in many papers. This paper deals with the behavior of rectangular Cesàro means at a point (x ₀, y ₀) for functions f(x, y) bounded in the square [?π; π]² and satisfying the condition |f(x ₀ + s, y ₀ + t) ? f(x ₀, y ₀)| ≤ ρ $ \left( {\sqrt {s^2 + t^2 } } \right) $ ^α , for some α ∈ (0, 1) and all s and t.     

On the approximation of conjugate functions from Holder classes by Fejér means

Получены асимптотич еские равенства для в еличин гдеr≧0 — целое, ω(t) — выпу клый модуль непрерыв ности и $$\bar \sigma _n (f;x) = - \frac{1}{\pi } \mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2}ctg\frac{t}{2} - \frac{1}{{4(n + 1)}}\frac{{\sin (n + 1)t}}{{\sin ^2 \tfrac{1}{2}t}}} \right)dt$$ сумма Фейера функцииf(х), сопряженной сf(x).