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.     

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.     

A theorem on the absolute Cesàro summability of factored fourier series

Summary In this paper, the author proves a theorem on the absoluteCesàro summability of factoredFourier series. His theorem extends a theorem ofMatsumoto and generalizes a theorem ofPrasad andBhatt.     

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,(+,+)]- .

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

The absolute generalized harmonic-Cesàro summability of a Fourier series

[8] . , , [2] , . - , , , . .     

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).     

Tauberian conditions for Cesàro summability of integrals

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

A remark about the statistical Cesàro summability and the Orlicz-Pettis theorem

We prove a version of the Orlicz-Pettìs theorem within the frame of the Statistical Cesàro summability.     

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:

Factorization of Cesàro and Hilbert matrices based on generalized Cesàro matrix

ABSTRACT

In this paper, we introduce factorizations for the Hilbert and Cesàro matrices based on generalized Cesàro matrix and as the application, we generalize the Hardy's inequality versus Hilbert's.     

Strong Cesàro summability of double Fourier integrals

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on . Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

On the Cesàro summability of orthogonal sequences

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

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.     

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.