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Double sequence core theorems. (English) Zbl 0949.40007

Let \(s=\{s_n\}\) denote a complex sequence. Define \(R_n\) to be the smallest closed convex region of the complex plane which contains \(\{s_n, s_{n+1}, \dots\}\), \(R:=\bigcap^\infty_{n=1} R_n\). The set \(R\) is called the core of \(s\). Knopp’s core theorem states that, if \(A\) is a nonnegative regular matrix, then the core of the \(A\)-transform of \(s\) is contained in the core of \(s\). For double sequences, the usual convergence used is that of Pringsheim, which leads to the Toeplitz conditions of regularity of a four-dimensional matrix (written RH-regular), for convergent sequences which are also bounded. In this paper the author defines the concept of a core sequence for double sequences and proves the two-dimensional analog of Knopp’s core theorem. The notation \(P\)-limit means the limit in the Pringsheim sense. The author also proves that, for all real-valued double sequences \([x]\), the condition \(P\)-\(\lim \sup[Ax] \leq P\)-\(\lim\sup [x]\) is equivalent to \(A\) being an RH-regular matrix satisfying \(P\)-\(\lim_{m,n} \sum^\infty_{k,j=0} |a_{mnjk} |=1\).

MSC:

40B05 Multiple sequences and series
40C05 Matrix methods for summability
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