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 $$\sigma_n^{T}(f)$$ of Vilenkin-Fourier series with respect to a bounded Vilenkin system. We give a sufficient condition for convergence for each integrable function $$f$$ , namely $$\sum_{k=1}^{n}t_{k,n}\to1\quad\text{and}\quad\sum_{k=1}^{n}t_{k,n}^{2}=O\left(\frac1n\right).$$ In comparison with previous results in this topic, we do not assume monotonicity of the defining sequence $$(t_{k,n},1\leq k\leq n, k\in\mathbb{P})$$ 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 $$\alpha>1/2$$ .