Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh–Fourier series

In the present paper, we prove the almost everywhere convergence and divergence of subsequences of Cesàro means with zero tending parameters of Walsh–Fourier series.

Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

We generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres

UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .

On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series

UDC 517.5 We establish approximation properties of Ces\`{a}ro means with α , β ϵ ( 0,1 ) of Vilenkin\,--\,Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means σ n , m - α , - β ( x , y , f ) to f ( x , y ) in the L p -metric.