Let \(\{a_{n}:\ n\in \mathbb {P}\}\) be an increasing sequence of positive integers. For every \(n\in \mathbb {P}\) let \(\{t_{k,a_{n}}: 1\le k\le a_{n},\ k\in \mathbb {P}\}\) be a finite sequence of non-negative numbers such that \(\begin{aligned} \sum _{k=1}^{a_{n}} t_{k,a_{n}}=1 \end{aligned}\) holds and \(\begin{aligned} \lim _{n\rightarrow \infty }t_{k,a_{n}}=0 \end{aligned}\) is satisfied for any fixed k. Our main result (Theorem 6.5) is that we prove \(L_{1}\) -norm convergence \(\begin{aligned} \sigma _{a_{n}}^{T}(f)\rightarrow f \end{aligned}\) (for example, but not limited to \(a_{n}:=2^{n}\) , see Corollary 6.6 and Sect. 7) with weaker conditions than it was known before for matrix transform means and for some special means, namely Nörlund and weighted ones.
