---
tag: YF1KAWB5MB
kind: concept
name: Quasi-analytic function
msc: ["26E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-analytic_function
same_as:
  wikidata: Q7269437
  wikipedia: Quasi-analytic function
---

## Summary

A quasi-analytic class is a class of [[smooth functions|0V33BNKZ9S]] in which a function is determined by its derivatives at a single point.
Real [[analytic functions|081WKEDK6C]] form such a class, but there are larger ones.
Quasi-analytic classes are defined by bounds on the growth of derivatives.
The [[Denjoy–Carleman theorem|KV0GW9K7JS]] says exactly which bounds give a quasi-analytic class.

## Explanations

### L1 human

For an analytic function, the derivatives at one point fix the function everywhere.
For general smooth functions this fails: a smooth function can vanish to all orders at a point and still be nonzero.
Quasi-analytic functions keep the good property of analytic functions without being analytic.
They are smooth functions whose derivatives do not grow too fast.