Difference between revisions of "Dini theorem"
From Encyclopedia of Mathematics
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| − | If the functions | + | If the functions $u_n$, $n=1,2,\ldots,$ are continuous and non-negative on a segment $[a,b]$ and if the sum of the series $\sum_{n=1}^\infty u_n$ is a continuous function on this segment, then the series converges uniformly on $[a,b]$. Dini's theorem can be generalized to the case when an arbitrary [[Compactum|compactum]] is the domain of definition of the functions $u_n$. |
Latest revision as of 13:25, 9 August 2014
on uniform convergence
If the functions $u_n$, $n=1,2,\ldots,$ are continuous and non-negative on a segment $[a,b]$ and if the sum of the series $\sum_{n=1}^\infty u_n$ is a continuous function on this segment, then the series converges uniformly on $[a,b]$. Dini's theorem can be generalized to the case when an arbitrary compactum is the domain of definition of the functions $u_n$.
How to Cite This Entry:
Dini theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini_theorem&oldid=32779
Dini theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini_theorem&oldid=32779
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article