Math 2.0

Real functions Definition

Quasi-analytic function

Builds on Smooth functionHigher derivativeAnalytic functionDenjoy–Carleman theoremSmooth non-analytic function

Definition

Let a<ba < b and let M=(Mk)k≥0M = (M_k)_{k \ge 0} be a sequence of positive real numbers. The class CM([a,b])C^M([a, b]) is the set of smooth functions f ⁣:[a,b]→ℝf \colon [a, b] \to \R for which there are constants A,B>0A, B > 0 with ∣f(k)(x)∣≤ABkMkfor all k≥0 and all x∈[a,b].\abs{f^{(k)}(x)} \le A B^k M_k \quad \text{for all } k \ge 0 \text{ and all } x \in [a, b]. The class CM([a,b])C^M([a, b]) is quasi-analytic if the following holds. If f∈CM([a,b])f \in C^M([a, b]) and f(k)(x0)=0f^{(k)}(x_0) = 0 for some x0∈[a,b]x_0 \in [a, b] and all k≥0k \ge 0, then f=0f = 0 on [a,b][a, b]. A quasi-analytic function is a function in a quasi-analytic class.

Remarks

Equivalently, two functions in a quasi-analytic class that have the same derivatives of all orders at one point are equal. At the endpoints, the derivatives are one-sided.

Example: Mk=k!M_k = k!. A function with ∣f(k)∣≤ABkk!\abs{f^{(k)}} \le A B^k k! on [a,b][a, b] is analytic, with Taylor series of radius at least 1/B1/B around each point. If all its derivatives vanish at x0x_0, then it vanishes near x0x_0, and by moving along [a,b][a, b] in steps it vanishes everywhere. So CM([a,b])C^M([a, b]) is quasi-analytic.

Non-example: Mk=(k!)2M_k = (k!)^2. Here ∑k≥1Mk−1/Mk=∑k≥11/k2\sum_{k \ge 1} M_{k-1}/M_k = \sum_{k \ge 1} 1/k^2 converges, so by the Denjoy–Carleman theorem the class is not quasi-analytic. It contains a nonzero function that vanishes on a subinterval, and all its derivatives vanish at the points of that subinterval.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Bump function, concept, no evidence levelBump functionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionQuasi-analytic function, concept, no evidence levelQuasi-analytic functi…conceptSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionAnalytic function, definition, no evidence levelAnalytic functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinitionDenjoy–Carleman theorem, statement, no evidence levelDenjoy–Carleman theor…statementSmooth non-analytic function, counterexample, no evidence levelSmooth non-analytic f…counterexampleQuasi-analytic function, definition, no evidence levelQuasi-analytic functi…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Bump function concept
  2. Limit point of a set definition
  3. Quasi-analytic function concept
  4. Sequence definition
  5. Limit of a function definition
  6. Real sequence definition
  7. Continuity at a point definition
  8. Derivative definition
  9. Series definition
  10. Continuous function definition
  11. Differentiable function definition
  12. Power series definition
  13. Derivative function definition
  14. Analytic function definition
  15. Higher derivative definition
  16. Taylor polynomial definition
  17. Continuously differentiable function definition
  18. Smooth function definition
  19. Taylor series definition
  20. Denjoy–Carleman theorem statement
  21. Smooth non-analytic function counterexample
  22. Quasi-analytic function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quasi-analytic_function, Definitions
About this page
Tag8M4N38VGEE
URIm2://8M4N38VGEE
Kinddefinition
WritingW0 stub
MSC26E10
DefinesQuasi-analytic function
UsesSmooth function, Higher derivative, Analytic function, Denjoy–Carleman theorem, Smooth non-analytic function
Changed2 November 2026

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