Norm Convergence and Divergence of Matrix Transform Means of Vilenkin-Fourier Series

We study the norm convergence of sequences of the general matrix transform means σₙT(f) of Vilenkin-Fourier series with respect to a bounded Vilenkin system. We give a sufficient condition for convergence for each integrable function f , namely ∑ₖ₌₁ⁿtk,n→1∑ₖ₌₁ⁿtk,n²=O(1n). In comparison with previous results in this topic, we do not assume monotonicity of the defining sequence (tk,n,1≤ k≤ n, k) of the matrix T . The T summation is a common generalisation of several well-known summation methods, such as Fejér, Weierstrass, Riesz, Picard, Bessel, and Nörlund summation methods. The method also includes Cesàro means; the present sufficient condition covers the range α>1/2 .