István Blahota

On a Norm Inequality with Respect to Vilenkin-Like Systems

We consider a new system introduced by G. Gát (see e.g. [3]). This is a common generalization of several well-known systems. We prove a norm inequality with respect to this system.

On the Nörlund means of Vilenkin-Fourier series

We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {qk: k ⩾ 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results.In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {qk: k ⩾ 0} in an appropriate way.

On the maximal value of Dirichlet and Fejér kernels with respect to the Vilenkin-like space

The concept of the Vilenkin-like system was introduced by G. Gat in [4]. This orthonormal system is a generalisation of several well-known ones, see the list of examples later. In the case of every cited system, domains of the systems’ functions were some special groups. The domain of the observed system in this paper is a set without any operation on it. Let m := (m0,m1, . . . ) denote by a sequence of positive integers not less than 2. Denote by Gmj a set, where the number of the elements is mj (j ∈ N). Define the measure on Gmk as follows

A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

Abstract The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p ≤ 1 {0<p\leq 1} . We also prove that the result is sharp in a particular sense.