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On double sequences of continuous functions having continuous P-limits. (English) Zbl 1289.40014

D. C. Gillespie and W. A. Hurwitz [Trans. Am. Math. Soc. 32, 527–543 (1930; JFM 56.0212.03)] proved the following result: If \(\left( s_{n}\right) \) is a sequence of functions continuous on a compact subset \(A\) of a metric space that converges pointwise to a continuous funtion \(s\) and is uniformly bounded on \(A,\) then there exists a regular nonnegative matrix \(T=\left( a_{nk}\right) \) (of a special type) such that \(\sum_{k}a_{nk}s_{k}\left( x\right) \rightarrow s\left(x\right) \) uniformly in \(x\in A\). The authors extend this result to double sequences of functions \(\left( s_{mn}\right) \) defined on \(A.\)

MSC:

40A30 Convergence and divergence of series and sequences of functions
40B05 Multiple sequences and series
40C05 Matrix methods for summability

Citations:

JFM 56.0212.03
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