General Summability Method Research Articles

Wiener amalgams, Hardy spaces and summability of Fourier series

AbstractA general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebraW(C,ℓ1)($\mathbb{R}$d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from theHpHardy space toLp(orHp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.

Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy–Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type ( 1 , 1 ) , provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σ T θ f → f over a cone-like set a.e. for all f ∈ L 1 ( R d ) . Moreover, σ T θ f ( x ) converges to f ( x ) over a cone-like set at each Lebesgue point of f ∈ L 1 ( R d ) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

Herz spaces and summability of Fourier transforms

AbstractA general summability method is considered for functions from Herz spaces Kαp,r (ℝd ). The boundedness of the Hardy–Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ ‐means σθ T f is also bounded on the corresponding Herz spaces and σθ T f → f a.e. for all f ∈ K–d /p p,∞ (ℝd ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (ℝd ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (ℝd ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Segal Algebra ${\bf S}_0({\Bbb R}^d)$ and Norm Summability of Fourier Series and Fourier Transforms

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L1-norm convergence of the θ-means σnθf to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger’s Segal algebra \({\bf S}_0({\Bbb R}^d)\), then these convergence results hold. Some new sufficient conditions are given for θ to be in \({\bf S}_0({\Bbb R}^d)\). A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case.

Mean Ergodicity and Mean Stability of Regularized Solution Families

We prove general theorems on mean ergodicity and mean stability of regularized solution families with respect to fairly general summability methods. They can be applied to integrated solution families, integrated semigroups and cosine functions. In particular, through applications with modified Cesaro, Abel, Gauss, and Gamma like summability methods we deduce particular results on mean ergodicity and mean stability of polynomially bounded C 0-semigroups and cosine operator functions.

-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallee-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

Marcinkiewicz- $$\theta$$ -Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function \(\theta\). Under some conditions on \(\theta\) we show that the maximal operator of the Marcinkiewicz-\(\theta\)-means of a tempered distribution is bounded from \(H_p \left( {R^2 } \right)\) to \(L_p \left( {R^2 } \right)\) for all \(p_0 < p \leqq \infty \) and, consequently, is of weak type \(\left( {1,1} \right)\), where \(p_0 < 1\) depends only on \(\theta \). As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-\(\theta \)-means of a function \(f \in L_1 \left( {R^2 } \right)\) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-\(\theta\)-means are uniformly bounded on the spaces \(H_p \left( {R^2 } \right)\) and so they converge in norm \(\left( {p_0 < p < \infty } \right)\). Some special cases of the Marcinkievicz-\(\theta \)-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, de la Vallee-Poussin, Rogosinski and Riesz summations.

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces Hp (T × … × T) (d = d1 + … + dkand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θj having integrable Fourier transforms. Under some conditions on θj we show that the maximal operator of the θ-means of a distribution is bounded from Hp (T × … × T) to Lp (Td) where p0 < p < ∞ and p0 < 1 is depending only on the functions θj. By an interpolation theorem we get that the maximal operator is also of weak type (L1) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L1. Moreover, we prove that the θ-means are uniformly bounded on the spaces Hp (T × … × T) whenever p0 <p < ∞, thus the θ-means converge to f in (T × … × T) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.

θ-Summation and Hardy Spaces

A general summability method of Fourier series and Fourier transforms is given with the help of an integrable function θ having integrable Fourier transform. Under some weak conditions on θ we show that the maximal operator of the θ-means of a distribution is bounded from Hp(T) to Lp(T) (p0<p<∞) and is of weak type (1,1), where Hp(T) is the classical Hardy space and p0<1 is depending only on θ. As a consequence we obtain that the θ-means of a function f∈L1(T) converge a.e. to f. For the endpoint p0 we get that the maximal operator is of weak type (Hp0(T), Lp0(T)). Moreover, we prove that the θ-means are uniformly bounded on the spaces Hp(T) whenever p0<p<∞ and are uniformly of weak type (Hp0(T), Hp0(T)). Thus, in the case f∈Hp(T), the θ-means converge to f in Hp(T) norm (p0<p<∞). The same results are proved for the conjugate θ-means and for Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.

Some theorems on the general summability methods

In this paper a new theorem which covers many methods of summability is proved. Several results are also deduced.

A Tauberian Theorem for the General Euler-Borel Summability Method

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m &gt; n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

Almost convergence and nonlinear ergodic theorems

The first ergodic theorems for nonlinear nonexpansive mappings and semigroups in Hilbert space were established by Baillon [I, 2, 31 and by Baillon and BrCzis [4]. BrCzis and Browder [6] have recently replaced the Cesaro method in the discrete case by more general summability methods. A similar extension in the continuous case has been obtained in [lo]. The purpose of this note is to show how Lorentz’s concept of almost convergence [9] can be used to improve these results and to simplify their proofs. Following Lorentz, we say that a regular matrix {a,,,<} is strongly regular

Nonlinear evolution equations and nonlinear ergodic theorems

Publisher Summary This chapter discusses nonlinear evolution equations and nonlinear ergodic theorems. It discusses certain aspects of the asymptotic behavior of generalized solutions of nonlinear evolution equations and of nonlinear nonexpansive semigroups in Banach spaces. The chapter discusses some results that can be used to derive iterative procedures for the construction of zeros of accretive sets. The first ergodic theorems for nonlinear nonexpansive mappings and semigroups in Hilbert space were established by Baillon and Brgzis. Bre'zis and Browder also simplified the original arguments in the discrete case and replaced the Ceslro method by more general summability methods.

On Riemann summability of functions

1. Let k be a positive integer. With the usual terminology, the serieswill be said to be Riemann summable (R, k) to l ifconverges for all sufficiently small x, and tends to l as x → 0. Here we take (sin nx)/(nx) as meaning 1 when n = 0. A more general summability method which has been considered by various authors ((1), (2), (5), (6), (8), (9), (10)), and is usually denoted by (ℜ, λ,k) is obtained by replacing (2) bywhere λ = {λn} is a sequence of non-negative numbers increasing to ∞.