General Summability Method Research Articles

On absolute weighted arithmetic mean summability of infinite series and Fourier series

Quite recently, we have obtained two main theorems dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series [C. R. Math. Acad. Sci. Paris 359 (2021), pp. 323–328]. In this paper, we have generalized these theorems for a general summability method. We have also obtained some new and known results for certain absolute summability methods.

Summability in anisotropic mixed-norm Hardy spaces

<abstract><p>Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.</p></abstract>

General Summability Methods in the Approximation by Bernstein–Chlodovsky Operators

In this paper, by using regular summability methods we modify the Bernstein–Chlodovsky operators in order get more general and powerful results than the classical aspects. We study Korovkin-type approximation theory on weighted spaces. As a special case, it is possible to Cesàro approximate (arithmetic mean convergence) to the test function although it fails for the classical Bernstein–Chlodovsky operators. At the end of the paper, we extend our results to the multi-dimensional case.

Summability of Fourier series in periodic Hardy spaces with variable exponent

Let $$p(\cdot ) \mathbb{T}\it ^n\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Holder condition and $$0<q \le \infty $$ . We introduce the periodic variable Hardy and Hardy–Lorentz spaces $$H_{p(\cdot )}(\mathbb{T}\it ^d)$$ and $$H_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ and prove their atomic decompositions. A general summability method, the so called $$\theta $$ -summability is considered for multi-dimensional Fourier series. Under some conditions on $$\theta $$ , it is proved that the maximal operator of the $$\theta $$ -means is bounded from $$H_{p(\cdot )}(\mathbb{T}\it ^d)$$ to $$L_{p(\cdot )}(\mathbb{T}\it ^d)$$ and from $$H_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ to $$L_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ . This implies some norm and almost everywhere convergence results for the summability means. The Riesz, Bochner–Riesz, Weierstrass, Picard and Bessel summations are investigated as special cases.

A new note on generalized absolute Cesàro summability factors

Quite recently, in [10], we have proved a theorem dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. In this paper, we generalize this theorem for the more general summability method. This new theorem also includes some new and known results.

Lebesgue points and restricted convergence of Fourier transforms and Fourier series

In this paper, a general summability method of multi-dimensional Fourier transforms, the so-called [Formula: see text]-summability, is investigated. It is shown that if [Formula: see text] is in a Herz space, then the summability means [Formula: see text] of a function [Formula: see text] converge to [Formula: see text] at each modified Lebesgue point, whenever [Formula: see text] and [Formula: see text] is in a cone. The same holds for Fourier series. Some special cases of the [Formula: see text]-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, Cesàro, de la Vallée-Poussin, Rogosinski and Riesz summations.

Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability

A general summability method of multi-dimensional Fourier transforms, the so called \({\theta}\)-summability is investigated. Under some conditions on \({\theta}\) we show that the Marcinkiewicz-\({\theta}\)-means of a function \({f\in W(L_1,\ell_\infty)({\mathbb{R}}^d)}\) converge to \({f}\) at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of \({f\in W(L_p,\ell_\infty)({\mathbb{R}}^d)}\), whenever \({1 < p < \infty}\). As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for \({f\in W(L_1,\ell_\infty)({\mathbb{R}})}\) and for strong \({\theta}\)-summability.

Some equivalence theorems on absolute summability methods

We obtained necessary and sufficient conditions for the equivalence of two general summability methods. Some new and known results are also obtained.

On certain double A-summability methods

The aim of this paper is to continue our investigations in line of our recent paper, Savas [24] and [26]. We introduce the notion of AI- double statistical convergence which includes the new summability methods studied in [24] and [23] as special cases and make certain observations on this new and more general summability method.

Lebesgue points of double Fourier series and strong summability

A general summability method of double Fourier series is given with the help of a double sequence θ(k,n). Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f∈L1(T2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so-called modified Lebesgue points of f∈Lp(T2), whenever 1<p<∞. The sufficient conditions of θ are proved for the Fejér, Abel and Cesàro summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong θ-summability. Some other special cases of the θ-summation are considered as well, such as the Weierstrass, Picard, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.

Lebesgue Points of Two-Dimensional Fourier Transforms and Strong Summability

We introduce the concept of modified strong Lebesgue points and show that almost every point is a modified strong Lebesgue point of \(f\) from the Wiener amalgam space \(W(L_1,\ell _\infty )({\mathbb {R}}^2)\). A general summability method of 2D Fourier transforms is given with the help of an integrable function \(\theta \). Under some conditions on \(\theta \) we show that the Marcinkiewicz-\(\theta \)-means of a function \(f\in W(L_1,\ell _\infty )({\mathbb {R}}^2)\) converge to \(f\) at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of \(f\in W(L_p,\ell _\infty )({\mathbb {R}}^2)\), whenever \(1<p<\infty \). As an application we generalize the classical 1D strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for \(f\in W(L_1,\ell _\infty )({\mathbb {R}})\) and for strong \(\theta \)-summability. Some special cases of the \(\theta \)-summation are considered, such as the Weierstrass, Abel, Picar, Bessel, Fejer, de La Vallee-Poussin, Rogosinski and Riesz summations.

A Tauberian Theorem for a General Summability Method

Abstract We investigate conditions under which Mϕ summability implies Abel summability and give the generalized Littlewood Tauberian theorem for Mϕ summability method.

Gabor Expansions and Restricted Summability

A general summability method, the so called θ-summability is considered for Gabor series. It is proved that the maximal operator of the θ-means defined in a cone is bounded from the local Hardy space hp to Lp and from the amalgam space W(hp,l∞) to W(LP,l∞). This implies the almost everywhere convergence of the θ-means for all f ∈ W(L1,l∞).

One-sided Tauberian conditions for a general summability method

Let ( u n ) be a sequence of real numbers and L be an additive summability method with some property. We show that if slow decrease of ( u n ) or one-sided boundedness of the classical control modulo of the oscillatory behavior of ( u n ) is a Tauberian condition for a general summability method L , then one-sided boundedness by a sequence with certain conditions of the general control modulo of the oscillatory behavior of integer order m of ( u n ) is also a Tauberian condition for L .

Summability of Gabor expansions and Hardy spaces

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for Gabor series. It is proved that the maximal operator of the θ-means is bounded from h p to L p and from the amalgam space W ( h p , ℓ ∞ ) to W ( L p , ℓ ∞ ) . This implies the almost everywhere convergence of the θ-means for all f ∈ W ( L 1 , ℓ ∞ ) .

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W(hp, l∞) to W(Lp, l∞). This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W(L1, l∞) ⊃ L1.

Formula omitted]-summability of higher-dimensional Fourier series

It is proved that the maximal operator of the ℓ 1 -Fejér means of a d -dimensional Fourier series is bounded from the periodic Hardy space H p ( T d ) to L p ( T d ) for all d / ( d + 1 ) < p ≤ ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the ℓ 1 -Fejér means of a function f ∈ L 1 ( T d ) converge a.e. to f . Moreover, we prove that the ℓ 1 -Fejér means are uniformly bounded on the spaces H p ( T d ) and so they converge in norm ( d / ( d + 1 ) < p < ∞ ) . Similar results are shown for conjugate functions and for a general summability method, called θ -summability. Some special cases of the ℓ 1 – θ -summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de la Vallée Poussin, Rogosinski and Riesz summations.

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space W ( h p □ , ℓ ∞ ) to W ( L p , ℓ ∞ ) . This implies the almost everywhere convergence of the θ-means for all f ∈ W ( L 1 , ℓ ∞ ) ⊃ L 1 .

Inequalities in summability theory of Fourier series

Some recent results on a general summability method, the so-called θ -summability, are summarized for one-dimensional Fourier series. Natural choices of θ are investigated, i.e., if θ is in Wiener amalgam spaces, Feichtinger's algebra or modulation spaces. Sufficient and necessary conditions are given for the uniform and L1 norm and a.e. convergence of the θ - means σ θ n f to the function f . The maximal operator of the θ -means is investigated and it is proved that it is bounded on Lp spaces and on Hardy spaces.

Pointwise Summability of Gabor Expansions

A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L1 or, more generally, when f∈W(L1,l∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.