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.
![{\displaystyle [a,b]\subset \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a659536067aaaac2db1c44613a09a715f0cf7246)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
Definitions
editLet be a sequence of positive real numbers with . Then we define the class of functions to be those which satisfy
![{\displaystyle C^{M}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92655a42dd04bd49accb77a74d6162efe55f0a6e)
![{\displaystyle f\in C^{\infty }([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646cf35fdca5efa0d2608e007c2025baf77b7521)
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
![{\displaystyle f\in C^{M}([a,b])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908b3aced663a8cc084c74a05dffa1655b26bb29)
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.
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
A function f is called a quasi-analytic function if f is in some quasi-analytic class.
References
edit- 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.