The Abel-type transformtions into \(G_w\) (Q1599776)

Let \(u= (u_n)^\infty_{n=0}\) be a number sequence. If the series \[ \sum^\infty_{k=0} {k+\alpha\choose k}\cdot u_k x^k\quad (\alpha> -1) \] is convergent for \(0< x< 1\) and \[ \lim(1- x)^{\alpha+ 1}\cdot \sum^\infty_{k=0} {k+\alpha\choose k}\cdot u_k x^k= L, \] then the sequence \(u= (u_n)\) is said to be \(A_\alpha\)-summable to \(L\); cf. \textit{D. Borwein} [On a scale of Abel-type summability methods, Proc. Camb. Phil. Soc. 53, 318-322 (1957; Zbl 0082.27602)]. The first author introduced a matrix analogon \(A_{\alpha,t}\) of method A putting \[ a_{nk}= {k+\alpha\choose k}\cdot t_n^k(1- t_n)^{\alpha+ 1}, \] where \(t= (t_n)\), \(0< t_n< 1\), \(\lim t_n= 1\), and \[ (A_{\alpha, t} u)_n= (1- t_n)^{\alpha+ 1} \sum^\infty_{k=0} {k+\alpha\choose k}\cdot u_k t^k_n. \] Put \(G_w= \{y= (y_k)\mid y_k= O(r^k)\) for some \(r\in (0,w)\) with \(0< w< 1\}\). Let \(A\) be a matrix. We set \(G_w(A)= \{y= (y_n)\mid Ay\in G_w\}\). The authors show that a matrix \(A_{\alpha,t}\) is a \(G_w\)-\(G_w\) matrix if and only if \((1- t)^{\alpha+ 1}\in G_w\). If \(-1<\alpha\leq 0\) and \(A_{\alpha, t}\) is a \(G_w\)-\(G_w\) matrix, then \(G_w(A_{\alpha, t})\) contains all sequences with bounded partial sums. A matrix \(A\) is said to be translative for \(u= (u_n)\) in \(G_w(A)\) if each of the sequences \(Tu\), \(Su\) is in \(G_w(A)\), where \(Tu= (u_1,u_2,\dots)\) and \(Su= (0,u_0, u_1,\dots)\). If \(-1<\alpha\leq 0\) then every \(G_w\)-\(G_w\) matrix \(A_{\alpha,t}\) is \(G_w\)-translative for each \(A_\alpha\)-summable sequence \(x\) in \(G_w(A_{\alpha, t})\).