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] Gát , G. , On the pointwise convergence of Cesàro means of two-variable functions with respect to unbounded Vilenkin systems , J. Approx. Theory , 128 ( 1 ) ( 2004 ), 69 – 99 . [6

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Abstract  

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 L 1-norm, only for certain values of α which depend on some parameter of the representative product system.

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the approximation properties of Cesàro means of negative order of Walsh–Fourier series , J. Approx. Theory , 115 ( 2002 ), no. 1 , 9 – 20 . [6] Goginava , U

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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 n,m /−α,−β (f; x, y) to f(x,y) in the L p-metric, p ∈ [1, ∞]. We also show that this sufficient condition cannot be improved.

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The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L p and C-modulus of continuity is studied.

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Abstract  

We prove the almost everywhere convergence of the Cesàro (C, α)-means of integrable functions σ n α ff for fL 1(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 [1]. 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 [2, 9].

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The behavior of generalized Cesro (C, α n)-means (α n ∈ (−1,0)) of trigonometric Fourier series of H ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.

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