\(r\)-convex sequences and matrix transformations
By considering a fixed chosen sequence \(B= (Bi)= (b_{np} (i))\) of infinite matrices, the authors introduce the notion of \(F_B\) convergence as follows. \(x\in \ell_\infty\) is said to be \(F_B\) convergent to the \(\text{Lim } Bx\) if \[ \lim_{n\to \infty} (B_i x)_n= \lim_{n\to \infty} \sum_{p=0}^\infty b_{np} (i) x_p= \text{Lim } Bx \] uniformly in \(i= 0, 1, 2, 3, \dots\). The authors obtain necessary and sufficient conditions on the matrix \(A= (a_{nk})\) to transform \(\text{SC}^r\) and \(\ell_1\) to \(F_B\) where \(\text{SC}^r\) is the set of all \(r\)-convex sequences and \(\ell_1= \{(x_n): \sum^\infty_{n=0} |x_n |< \infty\}\).
scientific article; zbMATH DE number 878107