User:Holmansf/Quasi-analytic classes

A quasi-analytic class of functions is a generalization of the class of analytic functions based upon the following fact. If f is an analytic function on an interval , and at some point f and all of its deriviates are zero, then f is identically zero on all of . Quasi-analytic classes are broader classes of functions for which this statement still holds true.

Definitions

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Let   be a sequence of positive real numbers with  . Then we define the class of functions   to be those   which satisfy

 

for some constant C and all non-negative integers k. If   this is exactly the class of real-analytic functions on  . The class   is said to be quasi-analytic if whenever   and

 

for some point   and all k, f is identically equal to zero. The Denjoy-Carleman theorem gives criteria on the sequence M under which   is a quasi-analytic class.

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

References

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  • Hörmander, Lars (1990). The Analysis of Linear Partial Differential Operators I. Springer-Verlag. ISBN 3-540-00662. {{cite book}}: Check |isbn= value: length (help)
  • Cohen, P. J. (1968). "A simple proof of the Denjoy-Carleman theorem". Amer. Math. Monthly. 75: 26–31.