Tauberian type gap conditions for Cesàro summation methods (Q1966214)

Let \(P\) and \(Q\) be summation methods for numerical series. A condition \(R\) on a sequence \(\{a_n\}\) is called a \(T_Q(P)\)-condition if any \(P\)-summable series \(\sum a_n\) such that \(\{a_n\}\) satisfies \(R\) is \(Q\)-summable. A sequence \(\{a_n\}\) belongs to \((G,k)\) if there exist a natural number \(C\), a real number \(q>1\), and a sequence of natural numbers \(\{n_r \}^\infty_{r=1}\) such that \[ {n_{r+1}\over n_r}\geq q\quad \text{for all }r,\quad a_n=0 \quad \text{if} \quad n\notin \bigcup^\infty_{r=1} (n_r-C,n_r], \] and for every \(r\) there are no more than \(k+1\) nonzero elements of \(\{a_n\}\) with indices \(n\in (n_r-C,n_r]\). The author proves the following interesting theorem: Let \(k\) be a fixed integer, and let \(\alpha\) be a fixed real number where \(\alpha> k\geq 0\). Then the condition \(\{a_n\}\in (G,k)\) is a \(T_{(C, k)} ((C,\alpha))\)-condition. He also recalls two theorems proved previously in his candidate thesis. These theorems show that the cited result is precise.