en>Richarddonkin: /* Determination of the Tikhonov factor */ Typo since Bayesian interpretation section is below, not above.

2015-01-07T13:38:47Z

Determination of the Tikhonov factor: Typo since Bayesian interpretation section is below, not above.

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en>Wile E. Heresiarch: /* Bayesian interpretation */ stable solution --> unique solution

2014-02-06T00:35:27Z

Bayesian interpretation: stable solution --> unique solution

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130.234.20.116: added category 'inverse problems'

2014-01-27T09:43:15Z

added category 'inverse problems'

← Older revision		Revision as of 11:43, 27 January 2014
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	~~Wilber Berryhill is what his wife enjoys~~ to ~~contact him~~ and ~~he totally loves this~~ title. ~~He is an information officer~~. ~~Her family members life in Ohio~~. ~~To play lacross~~ is the ~~thing I love most~~ of ~~all~~.<br><br>~~my blog post psychic phone readings~~ ([~~http~~://~~www~~.~~suedsteiermark~~.tv/~~users/RLoyola www~~.~~suedsteiermark.tv~~])		In [[mathematics]], the '''Dini''' and '''Dini-Lipschitz tests''' are highly precise tests that can be used to prove that the [[Fourier series]] of a [[function (mathematics)\|function]] converges at a given point. These tests are named after [[Ulisse Dini]] and [[Rudolf Lipschitz]].<ref>{{citation\|title=Introduction to Partial Differential Equations and Hilbert Space Methods\|author= Karl E. Gustafson\|year=1999\|publisher=Courier Dover Publications\|pages=121 \|url=http://books.google.com/?id=uu059Rj4x8oC&pg=PA121&dq=%22Dini+test%22\|isbn=978-0-486-61271-3}}</ref>

			== Definition ==

			Let ''f'' be a function on [0,2π], let ''t'' be some point and let δ be a positive number. We define the '''local modulus of continuity''' at the point ''t'' by

			:<math>\left.\right.\omega_f(\delta;t)=\max_{\|\varepsilon\| \le \delta} \|f(t)-f(t+\varepsilon)\|</math>

			Notice that we consider here ''f'' to be a periodic function, e.g. if ''t'' = 0 and ε is negative then we ''define'' ''f''(ε) = ''f''(2π + ε).

			The '''global modulus of continuity''' (or simply the [[modulus of continuity]]) is defined by

			:<math>\left.\right.\omega_f(\delta) = \max_t \omega_f(\delta;t)</math>

			With these definitions we may state the main results

			''Theorem (Dini's test): Assume a function f satisfies at a point t that''

			:<math>\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,d\delta < \infty.</math>

			''Then the Fourier series of f converges at t to f(t).''

			For example, the theorem holds with <math>\omega_f=\log^{-2}(\delta^{-1})</math> but does not hold with <math>\log^{-1}(\delta^{-1})</math>.

			''Theorem (the Dini-Lipschitz test): Assume a function f satisfies''

			:<math>\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.</math>

			''Then the Fourier series of f converges uniformly to f.''

			In particular, any function of a [[Hölder class]]{{clarify\|date=July 2010\|reason=Target of redirect does not define the term as used here}} satisfies the Dini-Lipschitz test.

			==Precision==

			Both tests are best of their kind. For the Dini-Lipschitz test, it is possible to construct a function ''f'' with its modulus of continuity satisfying the test with ''O'' instead of ''o'', i.e.

			:<math>\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.</math>

			and the Fourier series of ''f'' diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that

			:<math>\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,d\delta = \infty</math>

			there exists a function ''f'' such that

			:<math>\left.\right.\omega_f(\delta;0) < \Omega(\delta)</math>

			and the Fourier series of ''f'' diverges at 0.

			==See also==

			* [[Convergence of Fourier series]]
			* [[Dini continuity]]

			==References==
			{{reflist}}

			[[Category:Fourier series]]
			[[Category:Convergence tests]]

en>EmausBot: r2.7.2+) (Robot: Modifying tr:Ridge regresyon

2012-08-15T17:51:03Z

r2.7.2+) (Robot: Modifying tr:Ridge regresyon

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Wilber Berryhill is what his wife enjoys to contact him and he totally loves this title. He is an information officer. Her family members life in Ohio. To play lacross is the thing I love most of all.<br><br>my blog post psychic phone readings ([http://www.suedsteiermark.tv/users/RLoyola www.suedsteiermark.tv])

Tikhonov regularization - Revision history

en>Richarddonkin: /* Determination of the Tikhonov factor */ Typo since Bayesian interpretation section is below, not above.

en>Wile E. Heresiarch: /* Bayesian interpretation */ stable solution --> unique solution

130.234.20.116: added category 'inverse problems'

en>EmausBot: r2.7.2+) (Robot: Modifying tr:Ridge regresyon