A note on \(F_ B\)-convergence
Let S be the space of all complex sequences. For a sequence \(B=(B_ i)\) of infinite complex matrices let \(d_ B=\{x\in S\); \(Bx=((B_ ix))_ n\) exists\(\}\), where \(B_ i=(b_{np}(i))\), \(b_{np}(i)=1/(n+1)\), \(i\leq b\leq i+n\); 0, otherwise. If m denotes the Banach space of bounded sequences \(x=(x_ k)\), define \(F_ B\) by \(F_ B=\{x\in d_ B\cap m:\lim_{n}t_ n(i,x)\) exists uniformly in \(i\geq 0\), and is independent of \(i\}\), where \[ t_ n(i,x)=\sum^{\infty}_{p=0}b_{np}(i)x_ p,\;if\;n\geq 1\quad and\quad =\sum^{\infty}_{p=0}b_{op}(i)x_ p,\;if\;n=0, \] and \(b_{op}(i)=1\) if \(p=i\) and o otherwise. An infinite matrix A is called \((V-F_ B)\) matrix, if \(Ax\in F_ B\) whenever \(x\in V\), where V is the space of all sequences of bounded variation. In this article the following result is proved. Theorem: If A,B are as above such that \(_{n}\sum^{\infty}_{p=0}| b_{np}(i)| <\infty\) for every i, then A is a \((V-F_ B)\) matrix if and only if the following conditions hold. (i) \(_{p,k}| \sum^{\infty}_{\ell =k}a_{p\ell}| <\infty;\) (ii) There is a constant M such that \(\sup_{n}| \sum^{\infty}_{k=r}g_{nk}(i)| \leq M\), for \(i=0,1,2,...;\) (iii) \(\lim_{n}g_{nk}(i)=\alpha_ k\), uniformly for \(n=0,1,2,...\), and (iv) lim\(\sum^{\infty}_{k=0}g_{nk}(i)=\alpha\), uniformly for \(n=0,1,2,..\). where \(g_{nk}(i)=\sum^{\infty}_{p=0}b_{np}(i)a_{pk}\), for \(n\geq 1\), and \(\sum^{\infty}_{p=0}b_{op}(i)a_{pk}\), for \(n=0\).
scientific article; zbMATH DE number 4028218