Limitation theorems for absolute Nörlund summability

A limitation theorem concerning absolute Nörlund summability was proved recently. In the present paper limitation theorems concerning |N, p, q| summability are proved and corresponding theorems for |E, δ|, |C, α|, |C, α, β|, \(\left| {\bar N,p_n } \right|\) , |N, p _n ^α| methods are deduced as special cases.     

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.     

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.     

Triangular Fejér summability of two-dimensional Walsh-Fourier series

It is proved that the operators σ _n ^Δ of the triangular-Fejér-means of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space H _p to L _p for all 4/5 < p≤∞.     

Tauberian conditions for Cesàro summability of integrals

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

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 ¦.     

Fejér summability of multi-parameter bounded Ciesielski systems

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space H _p([0,1)^d ₁ × ¨ × [0,1)^d _l) to L _p([0,1)^d) if 1/2 < p < &infin; and m _j &ge; 0, |k _j| &le; m _j + 1. By an interpolation theorem, we get that the maximal operator is also of weak type (H ₁ ^#i,L ₁) (I=1,¨,l), where the Hardy space H ₁ ^#i is defined by a hybrid maximal function and H ₁ ^#i L(logL)^l-1. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f H ₁ ^#i and converge in a cone if f &isin; L ₁.     

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:

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.     

Strong summability theorems for H p (Δ d )

We prove that certain means of the (C,α,…,α)-means (α=1/p?1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space H _p to H _p (1≦p≦2). As a consequence we obtain strong summability theorems concerning (C,α,…,α)-means.     

An extension of Borel–Laplace methods and monomial summability

In this paper we will show that monomial summability can be characterized using Borel–Laplace like integral transformations depending of a parameter $0<s<1$. We will apply this result to prove 1-summability in a monomial of formal solutions of a family of partial differential equations.     

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.     

Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups

The (Nörlund) logarithmic means of the Fourier series is:

$t_n f = \frac{1}{{l_n }}\sum\limits_{k = 1}^{n - 1} {\frac{{S_k f}}{{n - k}}} , where l_n = \sum\limits_{k = 1}^{n - 1} {\frac{1}{k}} $

. In general, the Fejér (C,1) means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.

     

The absolute generalized harmonic-Cesàro summability of a Fourier series

[8] . , , [2] , . - , , , . .