Anisotropic diffusion - Revision history

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References

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	~~In mathematics~~, ~~a '''quasi-analytic''' class of '''functions''' is a generalization of the class of real [[analytic function]]s based upon~~ the ~~following~~ fact. If ''f'' is an analytic function on an interval [''a'',''b''] ⊂ '''R''', and at some point ''f'' and all of its derivatives are zero, then ''f'' is identically zero on all of [''a'',''b'']. Quasi-analytic classes are broader classes of functions for which this statement still holds true.		Hello there, I am Caprice. Montana has often been my dwelling spot. Due to the fact I was eighteen I have been doing the job as a [http://www.Ehow.com/search.html?s=generation generation] and distribution officer. It can be not a prevalent thing but what I like executing is playing crochet but I are not able to make it my career seriously. My spouse and I manage a site. You might want to check it out in this article: http://www.maderalegal.es/Files/doc/zapatillas-nike-air-max-baratas-673436.aspx<br><br>Also visit my webpage [http://www.maderalegal.es/Files/doc/zapatillas-nike-air-max-baratas-673436.aspx zapatillas nike air max baratas]

	~~==Definitions==~~

	~~Let <math>M=\{M_k\}_{k=0}^\infty</math> be a sequence of positive real numbers. Then we define~~ the ~~class of functions ''C''<sup>''M''</sup>([''~~a~~'',''b'']) to be those ''f'' ∈ ''C''<sup>∞</sup>(~~[~~''a'',''b'']) which satisfy~~

	:~~<math>\left \|\frac{d^kf}{dx^k}(x) \right \| \leq A^{k+1} M_k <~~/~~math>~~

	~~for all ''x'' ∈ [''a'',''b''], some constant ''A'', and all non-negative integers ''k''~~. ~~If ''M''<sub>''k''<~~/~~sub> ~~=~~ ''k''! this is exactly the class of real [[analytic function]]s on [''a'',''b''~~]. ~~The class ''C''<sup>''M''</sup>([''~~a~~'',''b''])~~ is ~~said~~ to ~~be ''quasi-analytic'' if whenever ''f'' ∈ ''C''<sup>''M''</sup>([''a'',''b''])~~ and

	~~:<math>\frac{d^k f}{dx^k}(x) = 0</math>~~

	~~for some point ''x'' ∈ [''~~a~~'',''b''] and all ''k'', ''f'' is identically equal~~ to ~~zero.~~

	~~A function ''f'' is called a ''quasi-analytic function'' if ''f'' is~~ in ~~some quasi-analytic class.~~

	~~==The Denjoy–Carleman theorem==~~

	~~The Denjoy–Carleman theorem, proved by {{harvtxt\|Carleman\|1926}} after {{harvtxt\|Denjoy\|1921}} gave some partial results, gives criteria on the sequence ''M'' under which ''C''<sup>''M''<~~/~~sup>([''a'',''b'']) is a quasi-analytic class~~. ~~It states that the following conditions are equivalent:~~
	*''C''<sup>''M''</sup>([''a'',''b'']) is quasi-analytic.
	*<math>\sum 1/~~L_j = \infty</math> where <math>L_j= \inf_{k\ge j}M_k^{1~~/~~k}<~~/~~math>.~~
	<math>\sum_j(M_j^)^{-~~1/j} = \infty</math>, where ''M''<sub>''j''</sub><sup>*</sup> is the largest log convex sequence bounded above by ''M''<sub>''j''</sub>~~.
	<~~math~~>~~\sum_jM_{j-1}^/M_j^* = \infty.~~<~~/math~~>

	~~The proof that the last two conditions are equivalent to the second uses~~ [~~[Carleman's inequality]].~~

	~~Example: {{harvtxt\|Denjoy\|1921}} pointed out that if ''M''<sub>''n''</sub> is given by one of the sequences~~
	~~:<math>n^n,\,(n\log n)^n,\,(n\log n\log \log n)^n,\,(n\log n\log \log n\log \log \log n)^n\dots</math>~~
	~~then the corresponding class is quasi-analytic. The first sequence gives analytic functions.~~

	~~==References==~~
	*{{citation\|authorlink=Torsten Carleman\|first= T. \|last=Carleman\|title=Les fonctions quasi-analytiques\|publisher= Gauthier-Villars \|year=1926}}
	*{{Citation \| doi=10.2307/2315100 \| authorlink=Paul Cohen (mathematician) \| last1=Cohen \| first1=Paul J. \| title=A simple proof of the Denjoy-Carleman theorem \| url=http://www.~~jstor~~.~~org~~/~~stable~~/~~2315100 \| id={{MathSciNet \| id = 0225957}} \| year=1968 \| journal=[[American Mathematical Monthly\|The American Mathematical Monthly]] \| issn=0002~~-~~9890 \| volume=75 \| pages=26–31 \| issue=1 \| publisher=Mathematical Association of America}}~~
	*{{citation\|authorlink=Arnaud Denjoy\|first= A. \|last=Denjoy\|title=Sur les fonctions quasi-~~analytiques de variable réelle\|journal= C.R. Acad. Sci. Paris \|volume= 173 \|year=1921\|pages= 1329–1331}}~~
	*{{Citation\|first=Lars\|last=[[Lars Hörmander\|Hörmander]]\|title=The Analysis of Linear Partial Differential Operators I\|publisher=Springer-~~Verlag\|year=1990\|isbn=3~~-~~540~~-~~00662}}~~
	*{{eom\|id=Q/q076370\|title=Quasi-analytic class\|first=A.~~F.\|last= Leont'ev}}~~
	*{{eom\|id=C/c020430\|title=Carleman theorem\|first=E.D.\|last= Solomentsev}}
	~~[[Category:Smooth functions]~~]

en>KYN: Undid revision 584406348 by 141.16.91.34 (talk) rv edit made without explanaition

2013-12-04T13:31:58Z

Undid revision 584406348 by 141.16.91.34 (talk) rv edit made without explanaition

← Older revision		Revision as of 15:31, 4 December 2013
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	~~They call~~ the ~~author Nathanial~~. ~~Playing handball may~~ be the ~~thing Vehicles most. Puerto Rico exactly where he~~'~~s been living. Auditing has been her employment~~ for a ~~short time~~ and it's ~~something she actually enjoy~~. Go to ~~my website to find out more: http~~:/~~/www~~.~~facethepeople~~.~~es/~~on/~~ugg~~-~~baratas~~-en-~~londres~~-67.~~html~~<br><br>~~Also visit my web page; [~~http://www.~~facethepeople~~.es/on/~~ugg~~-~~baratas~~-en-~~londres~~-67.~~html botas ugg baratas valencia~~]		In mathematics, a '''quasi-analytic''' class of '''functions''' is a generalization of the class of real [[analytic function]]s based upon the following fact. If ''f'' is an analytic function on an interval [''a'',''b''] ⊂ '''R''', and at some point ''f'' and all of its derivatives are zero, then ''f'' is identically zero on all of [''a'',''b'']. Quasi-analytic classes are broader classes of functions for which this statement still holds true.

			==Definitions==

			Let <math>M=\{M_k\}_{k=0}^\infty</math> be a sequence of positive real numbers. Then we define the class of functions ''C''<sup>''M''</sup>([''a'',''b'']) to be those ''f'' ∈ ''C''<sup>∞</sup>([''a'',''b'']) which satisfy

			:<math>\left \|\frac{d^kf}{dx^k}(x) \right \| \leq A^{k+1} M_k </math>

			for all ''x'' ∈ [''a'',''b''], some constant ''A'', and all non-negative integers ''k''. If ''M''<sub>''k''</sub> = ''k''! this is exactly the class of real [[analytic function]]s on [''a'',''b'']. The class ''C''<sup>''M''</sup>([''a'',''b'']) is said to be ''quasi-analytic'' if whenever ''f'' ∈ ''C''<sup>''M''</sup>([''a'',''b'']) and

			:<math>\frac{d^k f}{dx^k}(x) = 0</math>

			for some point ''x'' ∈ [''a'',''b''] and all ''k'', ''f'' is identically equal to zero.

			A function ''f'' is called a ''quasi-analytic function'' if ''f'' is in some quasi-analytic class.

			==The Denjoy–Carleman theorem==

			The Denjoy–Carleman theorem, proved by {{harvtxt\|Carleman\|1926}} after {{harvtxt\|Denjoy\|1921}} gave some partial results, gives criteria on the sequence ''M'' under which ''C''<sup>''M''</sup>([''a'',''b'']) is a quasi-analytic class. It states that the following conditions are equivalent:
			*''C''<sup>''M''</sup>([''a'',''b'']) is quasi-analytic.
			*<math>\sum 1/L_j = \infty</math> where <math>L_j= \inf_{k\ge j}M_k^{1/k}</math>.
			<math>\sum_j(M_j^)^{-1/j} = \infty</math>, where ''M''<sub>''j''</sub><sup>*</sup> is the largest log convex sequence bounded above by ''M''<sub>''j''</sub>.
			<math>\sum_jM_{j-1}^/M_j^* = \infty.</math>

			The proof that the last two conditions are equivalent to the second uses [[Carleman's inequality]].

			Example: {{harvtxt\|Denjoy\|1921}} pointed out that if ''M''<sub>''n''</sub> is given by one of the sequences
			:<math>n^n,\,(n\log n)^n,\,(n\log n\log \log n)^n,\,(n\log n\log \log n\log \log \log n)^n\dots</math>
			then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

			==References==
			*{{citation\|authorlink=Torsten Carleman\|first= T. \|last=Carleman\|title=Les fonctions quasi-analytiques\|publisher= Gauthier-Villars \|year=1926}}
			*{{Citation \| doi=10.2307/2315100 \| authorlink=Paul Cohen (mathematician) \| last1=Cohen \| first1=Paul J. \| title=A simple proof of the Denjoy-Carleman theorem \| url=http://www.jstor.org/stable/2315100 \| id={{MathSciNet \| id = 0225957}} \| year=1968 \| journal=[[American Mathematical Monthly\|The American Mathematical Monthly]] \| issn=0002-9890 \| volume=75 \| pages=26–31 \| issue=1 \| publisher=Mathematical Association of America}}
			*{{citation\|authorlink=Arnaud Denjoy\|first= A. \|last=Denjoy\|title=Sur les fonctions quasi-analytiques de variable réelle\|journal= C.R. Acad. Sci. Paris \|volume= 173 \|year=1921\|pages= 1329–1331}}
			*{{Citation\|first=Lars\|last=[[Lars Hörmander\|Hörmander]]\|title=The Analysis of Linear Partial Differential Operators I\|publisher=Springer-Verlag\|year=1990\|isbn=3-540-00662}}
			*{{eom\|id=Q/q076370\|title=Quasi-analytic class\|first=A.F.\|last= Leont'ev}}
			*{{eom\|id=C/c020430\|title=Carleman theorem\|first=E.D.\|last= Solomentsev}}
			[[Category:Smooth functions]]

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2012-08-31T01:58:32Z

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