en>Kfetzer97: The paper referenced uses sensor fusion to estimate attitude, not altitude.

2014-10-10T13:34:26Z

The paper referenced uses sensor fusion to estimate attitude, not altitude.

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	[[Image:Highly oscillatory function.png\|right\|frame\|The Riemann-Lebesgue lemma states that the integral of a function like the above is small. The integral will approach zero as the number of oscillations increases.]]		<br><br><br><br>The line moves rapidly. Dedicated servers are ideal for large companies, online applications or games which require more power and resources. You also have the choice on which type of reseller account you would like, there are 5 accounts to pick from, the aluminum, copper, silver, gold as well as diamond plan. Each of these packages is designed to give you the maximum amount of flexibility you need for a price you can afford. They have entire page dedicated to help you to make the best choice. Here's more information on Hostgator Coupon Codes ([http://www.test.anti-xolod.te.ua/?option=com_k2&view=itemlist&task=user&id=1937 http://www.test.anti-xolod.te.ua/]) look at our own site. The other important factor when picking out a host is the performance and quality which presents you velocity and keeps your website up and running. Great content spreads across the web and allows you to create a community. These companies also rarely offer website templates and other tools that can help novices get off to a running start. Comment, comment, comment- Just as people comment on your blog, do so on others. It has been in existence for more than a decade which ensures shoppers with the most effective web hosting service possible.<br><br><br><br>Hostgator Free website builder will you allow you to do that. Website website hosting is usually low cost in case you do it the best way. And the next year? The Cpanel is simple to comprehend and I am in a [http://Rt.com/search/everywhere/term/position/ position] to handle my websites without having to pay other people to do it for me.Hostgator is a company that provides things to look for, great deals as well as listens to their clients. You don't have to follow billions of people with the hope they'll reciprocate, either. Server administrators and resellers alike have numerous functions to work with helping them manage their several domains. Established in 2002 by Brent Oxley the HostGator is among the main web hosting providers with customers on the entire world. Renown as one of the top web hosting services; it's an online service that aids you in creating your own website for your business or any personal interest you might have or just for fun.

	~~In [[mathematics]], the '''Riemann–Lebesgue lemma''', named after [[Bernhard Riemann]] and [[Henri Lebesgue]], is of importance in [[harmonic analysis]] and [[asymptotic analysis]].~~

	~~The lemma says that the [[Fourier transform]] or [[Laplace transform]] of an [[Lp space\|''L''~~<~~sup~~>1<~~/sup~~> ~~function]] vanishes at infinity.~~

	~~== Statement ==~~
	~~If ''f'' is [[Lp space\|'''L'''~~<~~sup~~>1<~~/sup~~> ~~integrable]]~~ on ~~'''R'''<sup>''d''</sup>~~, ~~that is~~ to ~~say~~, if the ~~Lebesgue integral~~ of ~~\|''f''\|~~ is ~~finite, then~~ the ~~[[Fourier transform]]~~ of '~~'f'' satisfies~~

	~~:<math> \hat{f}~~(z):~~=\int_{\mathbb{R}^{d}} f(x) e^{~~-~~iz \cdot x}\,dx \rightarrow 0\text{ as } \|z\|\rightarrow \infty~~.</~~math>~~

	==~~Other versions~~==
	~~The Riemann–Lebesgue lemma holds in a variety of other situations~~.

	* If ''f'' is ''L''<sup>1</sup> integrable and supported on (0, ∞), then the Riemann–Lebesgue lemma also holds for the Laplace transform of ''f''. ~~That is,~~
	~~::<math>\int_0^\infty f(t) e^{~~-~~tz}\,dt \to 0<~~/~~math>~~
	~~:as \|''z''\| → ∞ within the half-plane Re(''z''~~)~~ ≥ 0~~.

	* A version holds for [[Fourier series]] as well: if ''f'' is ~~an integrable function on an interval, then~~ the ~~[[Fourier coefficient]]s of ''f'' tend to 0 as ''n'' → ±∞,~~
	~~::<math>\hat{f}_n \ \to \ 0~~ .~~</math>~~
	~~:This follows by extending ''f'' by zero outside~~ the ~~interval,~~ and ~~then applying the version of the lemma on the entire real line~~.

	~~==Applications==~~
	~~The Riemann–Lebesgue lemma~~ can ~~be used~~ to ~~prove the validity of asymptotic approximations for integrals~~. ~~Rigorous treatments of the [[method of steepest descent]] and the [[method of stationary phase]]~~, ~~amongst others~~, ~~are based~~ on ~~the Riemann–Lebesgue lemma~~.

	~~==Proof==~~

	~~We'll focus on the one dimensional case, the proof~~ in ~~higher dimensions is similar. Suppose first that ''f'' is~~ a ~~[[compact set\|compactly]] supported [[smooth function]]~~. ~~Then integration by parts in each variable yields~~

	:<~~math~~> ~~\left\| \int f(x) e^{-izx}dx\right\|=\left\|\int \frac{1}{iz} f'(x)e^{-izx}dx\right\| \leq \frac{1}{\|z\|}\int\|f'(x)\|dx \rightarrow 0 \mbox{ as } z\rightarrow\pm\infty.~~ <~~/math~~>

	~~If ''f'' is an arbitrary integrable function, it may be approximated in the ''L~~<~~sup~~>1<~~/sup~~>~~'' by a compactly supported smooth function ''g''. Pick such a ''g'' so~~ that ~~''\|\|f-g\|\|<sub>L<sup>1</sup></sub><ε''~~. ~~Then~~

	:<math> \limsup_{z\rightarrow\pm\infty} \|\hat{f}(z)\| \leq \limsup_{z\to\pm\infty} \left\|\int (f(x)-g(x))e^{-ixz}dx\right\| + \limsup_{z\rightarrow\pm\infty} \left\|\int g(x)e^{-ixz}dx\right\| \leq \varepsilon+0=\varepsilon,</math>
	~~and since this holds for any ''ε>0'',~~ the ~~theorem follows~~.

	''The ~~case of non-real t''.~~
	~~Assume first that ''f'' has~~ a ~~compact support on <math>(0,\infty)<~~/~~math> and that ''f'' is continuously differentiable~~.
	~~Denote the Fourier~~/~~Laplace transforms of ''f'' and <math>f'<~~/~~math> by ''F'' and ''G'', respectively.~~
	~~Then <math>F(t)=G(t)~~/t</~~math>~~, ~~hence <math>F(z)\rightarrow 0</math>~~ as ~~<math>\|~~t~~\|\rightarrow\infty</math>.~~
	~~Because the functions~~ of ~~this form are dense in <math>L^1(0,\infty)</math>,~~ the ~~same holds for every~~ '~~'f''.~~

	~~==References==~~
	*{{cite book \| author =[[Salomon Bochner\|Bochner S.]], ~~[[K~~. S. ~~Chandrasekharan\|Chandrasekharan K~~.~~]] \| title=Fourier Transforms \| publisher= Princeton University Press \| year=1949}}~~
	* {{mathworld\|urlname=Riemann-LebesgueLemma\|title=Riemann–Lebesgue Lemma}}

	~~{{DEFAULTSORT:Riemann-Lebesgue lemma}}~~
	~~[[Category:Asymptotic analysis]]~~
	~~[[Category:Harmonic analysis]]~~
	~~[[Category:Lemmas]]~~
	~~[[Category:Theorems~~ in ~~analysis]]~~
	~~[[Category:Theorems in harmonic analysis]]~~

en>Monkbot: /* Applications */Fix CS1 deprecated date parameter errors

2014-01-29T22:34:36Z

Applications: Fix CS1 deprecated date parameter errors

New page

[[Image:Highly oscillatory function.png|right|frame|The Riemann-Lebesgue lemma states that the integral of a function like the above is small. The integral will approach zero as the number of oscillations increases.]]

In [[mathematics]], the '''Riemann–Lebesgue lemma''', named after [[Bernhard Riemann]] and [[Henri Lebesgue]], is of importance in [[harmonic analysis]] and [[asymptotic analysis]].

The lemma says that the [[Fourier transform]] or [[Laplace transform]] of an [[Lp space|''L''1 function]] vanishes at infinity.

== Statement ==
If ''f'' is [[Lp space|'''L'''1 integrable]] on '''R'''''d'', that is to say, if the Lebesgue integral of |''f''| is finite, then the [[Fourier transform]] of ''f'' satisfies

:<math> \hat{f}(z):=\int_{\mathbb{R}^{d}} f(x) e^{-iz \cdot x}\,dx \rightarrow 0\text{ as } |z|\rightarrow \infty.</math>

==Other versions==
The Riemann–Lebesgue lemma holds in a variety of other situations.

* If ''f'' is ''L''1 integrable and supported on (0, ∞), then the Riemann–Lebesgue lemma also holds for the Laplace transform of ''f''. That is,
::<math>\int_0^\infty f(t) e^{-tz}\,dt \to 0</math>
:as |''z''| → ∞ within the half-plane Re(''z'') ≥ 0.

* A version holds for [[Fourier series]] as well: if ''f'' is an integrable function on an interval, then the [[Fourier coefficient]]s of ''f'' tend to 0 as ''n'' → ±∞,
::<math>\hat{f}_n \ \to \ 0 .</math>
:This follows by extending ''f'' by zero outside the interval, and then applying the version of the lemma on the entire real line.

==Applications==
The Riemann–Lebesgue lemma can be used to prove the validity of asymptotic approximations for integrals. Rigorous treatments of the [[method of steepest descent]] and the [[method of stationary phase]], amongst others, are based on the Riemann–Lebesgue lemma.

==Proof==

We'll focus on the one dimensional case, the proof in higher dimensions is similar. Suppose first that ''f'' is a [[compact set|compactly]] supported [[smooth function]]. Then integration by parts in each variable yields

:<math> \left| \int f(x) e^{-izx}dx\right|=\left|\int \frac{1}{iz} f'(x)e^{-izx}dx\right| \leq \frac{1}{|z|}\int|f'(x)|dx \rightarrow 0 \mbox{ as } z\rightarrow\pm\infty. </math>

If ''f'' is an arbitrary integrable function, it may be approximated in the ''L1'' by a compactly supported smooth function ''g''. Pick such a ''g'' so that ''||f-g||L1<ε''. Then

:<math> \limsup_{z\rightarrow\pm\infty} |\hat{f}(z)| \leq \limsup_{z\to\pm\infty} \left|\int (f(x)-g(x))e^{-ixz}dx\right| + \limsup_{z\rightarrow\pm\infty} \left|\int g(x)e^{-ixz}dx\right| \leq \varepsilon+0=\varepsilon,</math>
and since this holds for any ''ε>0'', the theorem follows.

''The case of non-real t''.
Assume first that ''f'' has a compact support on <math>(0,\infty)</math> and that ''f'' is continuously differentiable.
Denote the Fourier/Laplace transforms of ''f'' and <math>f'</math> by ''F'' and ''G'', respectively.
Then <math>F(t)=G(t)/t</math>, hence <math>F(z)\rightarrow 0</math> as <math>|t|\rightarrow\infty</math>.
Because the functions of this form are dense in <math>L^1(0,\infty)</math>, the same holds for every ''f''.

==References==
*{{cite book | author =[[Salomon Bochner|Bochner S.]], [[K. S. Chandrasekharan|Chandrasekharan K.]] | title=Fourier Transforms | publisher= Princeton University Press | year=1949}}
* {{mathworld|urlname=Riemann-LebesgueLemma|title=Riemann–Lebesgue Lemma}}

{{DEFAULTSORT:Riemann-Lebesgue lemma}}
[[Category:Asymptotic analysis]]
[[Category:Harmonic analysis]]
[[Category:Lemmas]]
[[Category:Theorems in analysis]]
[[Category:Theorems in harmonic analysis]]

Sensor fusion - Revision history

en>Kfetzer97: The paper referenced uses sensor fusion to estimate attitude, not altitude.

en>Monkbot: /* Applications */Fix CS1 deprecated date parameter errors