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On the Fejér means of bounded Ciesielski systems

Ferenc Weisz (2001)

Studia Mathematica

We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space H p to L p if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L₁ as n...

On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

Ferenc Móricz (1995)

Studia Mathematica

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces H ( 1 , 0 ) ( 2 ) , H ( 0 , 1 ) ( 2 ) , or H ( 1 , 1 ) ( 2 ) . We prove that the maximal Fejér operator is bounded from H ( 1 , 0 ) ( 2 ) or H ( 0 , 1 ) ( 2 ) into weak- L 1 ( 2 ) , and also bounded from H ( 1 , 1 ) ( 2 ) into L 1 ( 2 ) . These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces L 1 l o g + L ( 2 ) , L 1 ( l o g + L ) 2 ( 2 ) , and L μ ( 2 ) with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

On the rate of strong summability of double Fourier series

Włodzimierz Łenski, Maria Topolewska (1998)

Mathematica Bohemica

Estimates of the strong means of Marcinkiewicz type with the Cesaro means of negative order in one of the variables instead of square partial sums are obtained by characteristics constructed on the basis of moduli of continuity.

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