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'Barb\u{a}lat's lemma'
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| Title of object: |
Barb\u{a}lat's lemma |
| Canonical Name: |
BarbualatsLemma |
| Type: |
Theorem |
| Created on: |
2004-12-10 12:26:52 |
| Modified on: |
2005-03-05 12:12:01 |
| Classification: |
msc:26A06 |
| Synonyms: |
Barb\u{a}lat's lemma=Barbalat's lemma |
Revision comment (for changes between this and next version):
| Changes for correction #6453 ('spike function'). |
Preamble:
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Content:
\begin{lemma}[Barb\u{a}lat]
Let $f \colon (0,\infty) \to {\mathbb{R}}$ be Riemann integrable and uniformly continuous then
\begin{equation*}
\lim_{t \to \infty} f(t) = 0 .
\end{equation*}
\end{lemma}
Note that if $f$ is non-negative, then Riemann integrability is the same as being $L^1$ in the sense of Lebesgue, but if $f$ oscillates then the Lebesgue integral may not exist.
Further note that the uniform continuity is required to prevent sharp ``spikes'' that might prevent the limit from existing. For example suppose we add a spike of height 1 and area $2^{-n}$ at every integer. Then the function is $L^1$, but
$f(t)$ would not have a limit at infinity.
\begin{thebibliography}{9}
\bibitem{LoRy}
Hartmut Logemann, Eugene P.\@ Ryan.
\PMlinkescapetext{Asymptotic behaviour of nonlinear systems}.
\emph{The American Mathematical Monthly}, 111(10):864--889,
2004.
\end{thebibliography} |
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