Tauberian conditions for Cesàro summability of integrals

In this paper we generalize some classical type Tauberian theorems given for Cesàro summability of integrals.     

A Tauberian theorem for Cesàro summability of integrals

In this paper we give a proof of the generalized Littlewood Tauberian theorem for Cesàro summability of improper integrals.     

Rate of Cesàro summability of double orthogonal series

Unimprovable estimates (in the class of double orthogonal series) are obtained on the rate of almost everywhere summability of orthogonal expansions of square-integrable functions by Cesàro methods of positive order. The conditions imposed on the coefficients are of classical type.     

Properties of Cesàro means of double Fourier series

The pointwise behavior of partial sums and Cesàro means of trigonometric series were studied in many papers. This paper deals with the behavior of rectangular Cesàro means at a point (x ₀, y ₀) for functions f(x, y) bounded in the square [?π; π]² and satisfying the condition |f(x ₀ + s, y ₀ + t) ? f(x ₀, y ₀)| ≤ ρ $ \left( {\sqrt {s^2 + t^2 } } \right) $ ^α , for some α ∈ (0, 1) and all s and t.     

A high-indices theorem for strong Cesàro summability of double series

[4] , [C,(, )]_;- (, 0 1) s, [C,(+,+)]- s , 0 0< , (i) , ; > > 1 , (ii) ,>0; >=1 ,>1-1/, [C,(,)- [C,(+,+)]- .

---

---

This work was done while the author was a visiting researcher at the Steklov Mathematical Institute, Moscow, U.S.S.R.     

On the Cesàro summability of orthogonal sequences

(C, ). , . 0<<1. 1) - ( _k ), _k =a _k , (C, ), . 2) , , (C, ) ; _k = =¦a _k ¦.     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).     

Cesàro summability of one- and two-dimensional Walsh-Fourier series

ВВОДьтсьp-кВАжИлОкАл ьНыЕ ОпЕРАтОРы И ОДНО МЕРНыЕ ДИАДИЧЕскИЕ МАРтИНг АльНыЕ пРОстРАНстВА хАРДИH _p . ДОкАжАНО, ЧтО ЕслИ сУБлИНЕИНыИ ОпЕРАтО РT p-кВАжИлОкАлЕН И ОгРА НИЧЕН ИжL _∞ ВL _∞, тО ОН ьВльЕтсь тАкжЕ ОгРАН ИЧЕННыМ ИжH _p ВL _p , (0<p<1). В кАЧЕстВЕ пРИ лОжЕНИь ДОкАжАНО, ЧтО МАксИМАльНыИ ОпЕРАт ОР ОДНОгО ЧЕжАРОВскОгО пАРАМЕтРА И МОДИФИцИ РОВАННых ЧЕжАРОВскИх сРЕДНИх МАРтИНгАлА ьВльЕтсь ОгРАНИЧЕННыМ ИжH _p ВL _p И ИМЕЕт слАБыИ тИп (L ₁,L ₁). Мы ВВОДИМ ДВУМЕРНыИ ДИА ДИЧЕскИИ гИБРИД пРОс тРАНстВ хАРДИH ₁ И пОкАжыВАЕМ, Ч тО МАксИМАльНыИ ОпЕРАт ОР сРЕДНИх ЧЕжАРО ДВУ МЕРНОИ ФУНкцИИ ИМЕЕт слАБыИ тИп (H ₁ ^# ,L ₁). тАк Мы пОлУЧАЕМ, Ч тО ДВУпАРАМЕтРИЧЕск ИЕ сРЕДНИЕ ЧЕжАРО ФУНкц ИИf ?H ₁ ^# ?L logL схОДьтсь пОЧтИ ВсУДУ к ИсхОДНОИ ФУНк цИИ.     

On the Cesàro summability of the ultraspherical series

Summary The author has obtained theorems for Cesàro summability of the ultraspherical series which are analogous to those of Izumi and Sunouchi (2) in Fourier series.     

On weighted uniform Cesàro summability of Jacobi-Fourier series

The aim of this paper is to give weighted function spaces in which the sequence of Cesàro means of the Jacobi-Fourier series are uniformly convergent. Error estimate for the approximation will also be considered.     

Factors for generalized absolute Cesàro summability

In this paper, a known theorem dealing with ${|C,\alpha |}_k$ summability factors has been generalized for ${|C,\alpha ,\beta |}_k$ summability factors.     

Singular integrals in the Cesàro sense

The existence of the singular integral ∫K(x, y)f(y)dy associated to a Calderón-Zygmund kernel where the integral is understood in the principal value sense TF(x)=lim_ε→0+∫_|x−y|>εK(x, y)f(y)dy has been well studied. In this paper we study the existence of the above integral in the Cesàro-α sense. More precisely, we study the existence of

Triangular Ces??ro summability of two dimensional Fourier series

It is proved that the maximal operator of the triangular Ces??ro means of a two-dimensional Fourier series is bounded from the periodic Hardy space $H_{p}(\mathbb{T}^{2})$ to $L_{p}(\mathbb{T}^{2})$ for all 2/(2+??)<p?Q?? and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Ces??ro means of a function $f \in L_{1}(\mathbb{T}^{2})$ converge a.e. to?f.     

Negative order Cesàro means of double Fourier series and bounded generalized variation

Under study is the convergence of the negative order Cesàro means of the double trigonometric Fourier series of functions of bounded generalized Λ-variation.     

Some Tauberian conditions for Cesàro summability method

Let u = (u _n ) be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that (u _n ) is slowly oscillating if the sequence of Cesàro means of (ω _n ^(m−1)(u)) is increasing and the following two conditions are hold: