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Zbl 0847.40004
Ahmad, Z.U.; Mursaleen; Khan, Q.A.
$r$-convex sequences and matrix transformations. (English)
[J] Acta Math. Hung. 69, No.4, 327-335 (1995). ISSN 0236-5294; ISSN 1588-2632/e

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 $\ell_1= \{(x_n): \sum^\infty_{n=0} |x_n |< \infty\}$.
[D.Somasundaram (Madurai)]

MSC 2000:: *40C05 Matrix methods in summability
46A45 Sequence spaces
26A51 Convexity, generalizations (one real variable)

Keywords: matrix transformations; convergence; $r$-convex sequences

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