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.     

Limitation theorems for absolute Nörlund summability

A limitation theorem concerning absolute Nörlund summability was proved recently. In the present paper limitation theorems concerning |N, p, q| summability are proved and corresponding theorems for |E, δ|, |C, α|, |C, α, β|, \(\left| {\bar N,p_n } \right|\) , |N, p _n ^α| methods are deduced as special cases.     

On the Nörlund means of Vilenkin-Fourier series

We prove and discuss some new (H _p ,L _p )-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q _k : k ? 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results.     

Nörlund summability of Fourier series and its conjugate series

Summary In this paper the author has obtained two theorems for the N?rlund summability of Fourier series and its conjugate series respectively under very general conditions.     

A study of double Fourier series by Nörlund summability

A theorem on the Nörlund summability of double Fourier series has been established; it generalizes various known results.     

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 by Nörlund means of double fourier series to continuous functions in two variables

We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions \(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and \(\tilde f^{(1,1)} (x,y)\) .     

Nörlund and Riesz mean of sequences of fuzzy real numbers

In this article we study some properties of the Nörlund and Riesz mean of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for the Nörlund and Riesz means to transform convergent sequences of fuzzy numbers into convergent sequences of fuzzy numbers with limit preserving.     

Almost Everywhere Convergence of Strong Nörlund Logarithmic Means of Walsh-Fourier Series

In this paper we study the maximal operator for a class of subsequences of strong Nörlund logarithmic means of Walsh-Fourier series. For such a class we prove the almost everywhere strong summability for every integrable function f.     

Some remarks on localization of Schrödinger means

We study localization and localization almost everywhere of Schrödinger means of functions in Sobolev spaces.     

On the Nörlund logarithmic means with respect to Vilenkin system in the martingale Hardy space <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>

We prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space H ₁.     

On Absolute Nörlund Spaces and Matrix Operators

In a more recent paper, the second author has introduced a space |C_α|_k as the set of all series by absolute summable using Ces`aro matrix of order α -1. In the present paper we extend it to the absolute N?rlund space |N_p~θ|_k taking N?rlund matrix in place of Ces`aro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.     

Hörmander-type multipliers on locally compact Vilenkin groups: theL 1(G)-case

In [3] and [4]Kitada presented Hörmander-type multiplier theorems for Lebesgue and Hardy spaces defined over a locally compact Vilenkin groupG. Like in the classical case, multipliers for the spaceL ¹(G) were not included in these results. In the present paper we discuss this particular case and we show how we need to modify the usual Hörmander multiplier condition to obtainL ¹ (G)-multipliers.