|
|
Line 1: |
Line 1: |
| 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]]
| |
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 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
Also visit my webpage zapatillas nike air max baratas