Barbălat’s lemma
Lemma (Barbălat).
Let $f\mathrm{:}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{\infty}\mathrm{)}\mathrm{\to}\mathrm{R}$ be Riemann integrable^{} and uniformly continuous^{} then
$$\underset{t\to \mathrm{\infty}}{lim}f(t)=0.$$ |
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 continuous^{} and ${L}^{1}$ (and thus Riemann integrable), but $f(t)$ would not have a limit at infinity.
References
- 1 Hartmut Logemann, Eugene P. Ryan. . The American Mathematical Monthly, 111(10):864–889, 2004.
Title | Barbălat’s lemma |
---|---|
Canonical name | BarbualatsLemma |
Date of creation | 2013-03-22 14:52:31 |
Last modified on | 2013-03-22 14:52:31 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 7 |
Author | jirka (4157) |
Entry type | Theorem |
Classification | msc 26A06 |
Synonym | Barbalat’s lemma |