Strong Summability Research Articles

On the strong summability of the Fourier–Walsh series in the Besov space

UDC 517.5 The Fourier–Walsh series of even continuous functions may be divergent at some points. Moreover, among integrable functions, there are functions such that their Fourier–Walsh series diverge everywhere on [ 0,1 ) . In this connection, it becomes necessary to consider various summation methods that would allow us to restore the function according to its Fourier–Walsh series. We also investigate the Besov space on a dyadic group in terms of strong summability. Finally, we present necessary information about the Fourier–Walsh transform.

Сильная аппроксимация функций двойными рядами Фурье в обобщённых гёльдеровых пространствах

We study some problems of strong approximation of the functions of double Fourier series in the generalized Holder spases. Order-accurate estimates for the strong summability functionals of double Fourier series are established, as wel as their approximation properties in the indicated spases.

Strong summability of double Vilenkin–Fourier series

In this paper we study the a.e. strong summability of the quadratical and rectangular partial sums of the two-dimensional Vilenkin–Fourier series for every two-dimensional functions belonging to L log L. We also study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Vilenkin–Fourier series.

Strong summability of double Vilenkin–Fourier series

In this paper we study the a.e. strong summability of the quadratical and rectangular partial sums of the two-dimensional Vilenkin–Fourier series for every two-dimensional functions belonging to L log L. We also study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Vilenkin–Fourier series.

Absolute and strong summability in degree $p \geqslant 1$ of series, associated with Fourier series, by matrix methods

We establish conditions of $|\gamma|_p$- and $[\gamma]_p$-summability in degree $p \geqslant 1$ of series, associated with Fourier series, at the point where $\gamma = \| \gamma_{nk} \|$ is the matrix of transformation of series to sequence.

Absolute and strong summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series and $r$ times differentiated conjugate Fourier series by matrix methods

We establish conditions of $|\gamma|_p$- and $[\gamma]_p$-summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series at the point where $\gamma = \| \gamma_{nk} \|$ is the matrix of transformation of series to sequence. Analogous conditions are considered also for $r$ times differentiated conjugate Fourier series.

Tauberian theorems in the case of strong summability in degree $p$ of double series

We establish a Tauberian theorem in the case of strong summability in degree $p$ of double series by matrix methods, give its application to Abel methods.

Strong summability of differentiated conjugate double Fourier series

Strong summability of differentiated conjugate double Fourier series.

Pointwise (H, Φ) Strong Approximation by Fourier Series of LΨ Integrable Functions

We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H, Φ) as an approximation version of the Totik type generaliza- tion of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from LΨ. As a measure of such approximation we will use the function constructed by function Ψ com- plementary to Φ on the base of definition of the LΨ points. Some corollary and remarks will also be given.

On points of strong summability of trigonometric integrals

On points of strong summability of trigonometric integrals

Strong Riesz summability of Fourier series

The notion of strong summability was introduced by Fekete (Math. És Termesz Ertesitö, 34 (1916), 759-786). Dealing with Nörlund summability of Fourier series Mittal (J. Math. Anal. Appl. 314 (2006), 75-84) has established a result on strong summability. We have established a new result on sufficient condition for strong Riesz summability of Fourier series.

Pointwise Strong Summability of Vilenkin–Fourier Series

In this paper, we give a description of points at which the strong means of VilenkinFourier series converge.

Almost Everywhere Strong C,1,0 Summability of 2-Dimensional Trigonometric Fourier Series

In this paper we study the a. e. exponential strong (C, 1, 0) summability of of the 2-dimensional trigonometric Fourier series of the functions belonging to L (log+L)2.

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

Abstract In this paper, we investigate the strong summability of two-dimensional Walsh–Fourier series obtained in [F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series, Studia Math. 117 1996, 2, 173–194] (see Theorem W) and prove the sharpness of this result.

Strong Summability of Two-Dimensional Vilenkin–Fourier Series

We study the exponential uniform strong summability of two-dimensional Vilenkin–Fourier series. In particular, it is proved that the two-dimensional Vilenkin–Fourier series of a continuous function f is uniformly strongly summable to a function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

Riesz means of Fourier series and integrals: Strong summability at the critical index

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index ( d − 1 ) / 2 (d-1)/2 may fail for functions in the Hardy space h 1 ( T d ) h^1(\mathbb T^d) , we prove sharp positive results for strong summability almost everywhere. For functions in L p ( T d ) L^p(\mathbb T^d) , 1 > p > 2 1>p>2 , we consider Riesz means at the critical index d ( 1 / p − 1 / 2 ) − 1 / 2 d(1/p-1/2)-1/2 and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces H p ( R d ) H^p(\mathbb {R}^d) for 0 > p > 1 0>p>1 .

Linear Functionals Connected with Strong Double Cesaro Summability

D. Borwein characterized linear functionals on the normed linear spaces wp and Wp. In this paper we extend his results by presenting definitions for the double strong Cesaro mean. Using these new notions of strongly p-Cesaro summable double sequence and strongly p-Cesaro summable bivariate function we present extensions of D. Borwein’s results.

λ-double statistical convergence of functions

In this paper, the definitions of ?-double strong summability and ?-double statistical convergence for real valued measurable functions of two variables defined on (1,?)x(1,?) are presented. Using these definitions we present a series of basic results. Additionally, inclusion theorems, extension of existing results in the literature, and their variations have been established.

On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series

In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL.

Pointwise left( H,Phi right) Strong Approximation by Fourier Series of L^{1} Integrable Functions

We will present an estimation of the generalized strong mean left( H,Phi right) as an approximation version of the Totik type generalization of the results of J. Marcinkiewicz and A. Zygmund on strong summability of Fourier series of integrable functions. As a measure of such approximation we will use the function constructed by function Psi complementary to Phi on the base of definition of the Gabisoniya points. Some corollary and remark will also be given.