Approximation by $T$ means with respect to Vilenkin system in Lebesgue spaces and Lipschitz classes

In this paper we present and prove some new results concerning approximation properties of $T$ means with respect to the Vilenkin system in Lebesgue spaces and Lipschitz classes for any $1\le p<\mathrm{\infty }$. As applications, we obtain extension of some known approximation inequalities.