A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².     

A note on absolute summability factors

In this paper, by using an almost increasing and δ-quasi-monotone sequence, a general theorem on φ- | C, α | _k summability factors, which generalizes a result of Bor [3] on φ |C, 1| _k summability factors, has been proved under weaker and more general conditions.     

A note on absolute summability factors

Summary We obtain sufficient conditions for the series <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\sum a_{n}\lambda_{n}$ to be absolutely summable of order $k$ by a<span lang=FR style='font-size:10.0pt; mso-ansi-language:FR'>triangular matrix.     

A note on two absolute summability methods

In this paper we have established a relation between |R,p _n ;δ|_k and |R,q _n ;δ|_k summability methods which generalizes the results of Bor [1] and Bosanquet [2].     

A new note on absolute summability factors

In this paper, a recent result dealing with  \(| \bar{N},p_{n}|_{k}\) summability factors, has been generalized for \(|\bar{N},p_{n},\theta_{n}|_{k}\) summability factors. Some new results have also been obtained.     

级数的常规可和，Cesàro可和与Abel可和的几点讨论

讨论级数常规可和、Cesaro可和与Abel可和的关系．利用数学分析级数理论,证明Abel可和适用范围最广,Cesaro可和其次,级数常规可和适用范围最小．这个结论丰富了经典级数理论,为实际应用中选用合适可和提供依据．     

A note on summability of multiple Laguerre expansions

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an -radial function is shown to be an -radial function that coincides with the Laguerre polynomial expansion of , which allows us to settle the problem of summability below the critical index for the -radial functions.

     

A note on absolute weighted mean summability factors

In this paper we have proved a main theorem concerning the | $\bar N$ , p _n; δ |_k summability methods, which generalizes a result of Bor and Özarslan [3].     

A note on strong matrix summability via ideals

Following the line of recent works of Das et al. (2011) [15] and Savas and Das (2011) [19] we apply the notion of ideals to strong matrix summability, and in this note, we introduce the notion of strong A^I-summability with respect to an Orlicz function. We make certain investigations on the classes of sequences arising out of this new summability method.     

A note on the Valiron method of summability

Acta Mathematica Hungarica -     

Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

Absolute summability of orthogonal series by Hausdorff methods

Sufficient conditions are obtained for the absolute summability of orthogonal series by the Hausdorff methods.Translated from Matematicheskie Zametki, Vol.10, No. 5, pp. 485–492, November, 1971.I should like to thank A. V. Efimov, whose helpful comments improved this paper.     

Convergence and summability of Fourier series

In this note we study (C, –) summability, 0<1, of Fourier series and their conjugates in the sense of convergence with respect to the L^P norm, 1p+.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 887–898, June, 1976.     

A note on Clebsch-Gordan series

The purpose of this note is to make explicit some of the implicit conjectures and questions in Bacławski, Adv. in Appl. Math. 5 (1984), 416–432. While one of the conjectures can be stated entirely in combinatorial terms, the others involve a mixture of combinatorics, algebra, and Lie theory.     

A note on orthonormal series

Theorems are proved on the strong summability of orthonormal series and an error in a recent paper of Sunouchi is corrected.