The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L1-norm, only for certain values of α which depend on some parameter of the representative product system.
We investigate approximation properties of Cesàro (C; −α, −β)-means of double Walsh-Fourier series with α, β ∈ (0, 1). As an application, we obtain a sufficient condition for the convergence of the means σ
(f; x, y) to f(x,y) in the Lp-metric, p ∈ [1, ∞]. We also show that this sufficient condition cannot be improved.
We prove the almost everywhere convergence of the Cesàro (C, α)-means of integrable functions σnαf → f for f ∈ L1(I), where I is the group of 2-adic integers for every α > 0. This theorem for the case of α = 1 was proved by the author . For the case of the (C, 1) Fejér means there are several generalizations known with respect to some orthonormal systems. One could mention the papers
The behavior of generalized Cesro (C, αn)-means (αn ∈ (−1,0)) of trigonometric Fourier series of Hω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.