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The dual spaces of sets of difference sequences of order $m$ and matrix transformations. (English) Zbl 1123.46007
Summary: Let $p={\left({p}_{k}\right)}_{k=0}^{\infty }$ be a bounded sequence of positive reals, $m\in ℕ$, and $u$ be s sequence of nonzero terms. If $x={\left({x}_{k}\right)}_{k=0}^{\infty }$ is any sequence of complex numbers, we write ${{\Delta }}^{\left(m\right)}x$ for the sequence of the $m$-th order differences of $x$ and ${{\Delta }}_{u}^{\left(m\right)}X=\left\{x={\left(x\right)}_{k=0}^{\infty }:u{{\Delta }}^{\left(m\right)}x\in X\right\}$ for any set $X$ of sequences. We determine the $\alpha$-, $\beta$- and $\gamma$-duals of the sets ${{\Delta }}_{u}^{\left(m\right)}X$ for $X={c}_{0}\left(p\right),c\left(p\right),{\ell }_{\infty }\left(p\right)$ and characterize some matrix transformations between these spaces ${{\Delta }}^{\left(m\right)}X$.
##### MSC:
 46A45 Sequence spaces 40H05 Functional analytic methods in summability
##### Keywords:
difference sequences; dual spaces; matrix transformations
##### References:
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