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The dual spaces of sets of difference sequences of order $m$ and matrix transformations. (English) Zbl 1123.46007
Summary: Let $p = (p_k)^{\infty }_{k = 0}$ be a bounded sequence of positive reals, $m\in \Bbb N$, and $u$ be s sequence of nonzero terms. If $x =(x_{k})^{\infty }_{k = 0}$ is any sequence of complex numbers, we write $\Delta ^{(m)} x$ for the sequence of the $m$-th order differences of $x$ and $\Delta ^{( m)}_{u} X = \{x = (x)^{\infty }_{k = 0} :u\Delta ^{(m)} x \in X\}$ for any set $X$ of sequences. We determine the $\alpha$-, $\beta$- and $\gamma$-duals of the sets $\Delta ^{(m)}_{u} X$ for $X = c_{0}(p), c(p), \ell _\infty (p)$ and characterize some matrix transformations between these spaces $\Delta ^{(m)} X$.

46A45Sequence spaces
40H05Functional analytic methods in summability
Full Text: DOI
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