Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .

Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems

UDC 517.5 We prove some new strong convergence theorems for partial sums and Fej\'er means with respect to the Vilenkin system.

Direct and strong converse inequalities for approximation with Fejér means

We present upper and lower estimates of the error of approximation of periodic functions by Fejér means in the Lebesgue spaces {L}_{2{\pi }}^{p}. The estimates are given in terms of a K-functional for 1\le p\le \infty and in terms of the first modulus of continuity in the case 1\lt p\lt \infty . We pay attention to the involved constants.

Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.