Math 2.0

Real functions Differentiation Theorem

Denjoy–Carleman theorem

Builds on Quasi-analytic functionSmooth functionAnalytic functionBump function

Statement

Let a<ba < b and let M=(Mk)k≥0M = (M_k)_{k \ge 0} be a sequence of positive real numbers with M0=1M_0 = 1. Assume that MM is logarithmically convex: Mk2≤Mk−1Mk+1M_k^2 \le M_{k-1} M_{k+1} for all k≥1k \ge 1. Let CM([a,b])C^M([a, b]) be the class of smooth functions ff on [a,b][a, b] with ∣f(k)∣≤ABkMk\abs{f^{(k)}} \le A B^k M_k on [a,b][a, b] for some constants A,B>0A, B > 0 and all k≥0k \ge 0. Then the following three conditions are equivalent.

  1. The class CM([a,b])C^M([a, b]) is quasi-analytic.
  2. ∑k=1∞Mk−1/k=∞\sum_{k=1}^{\infty} M_k^{-1/k} = \infty.
  3. ∑k=1∞Mk−1Mk=∞\sum_{k=1}^{\infty} \dfrac{M_{k-1}}{M_k} = \infty.

Remarks

When the series converge, one builds a nonzero function of the class that vanishes outside a small interval. A standard construction is an infinite convolution of normalized indicator functions of short intervals, whose lengths are related to the ratios Mk−1/MkM_{k-1}/M_k. Carleman's proof of the direction from divergence to quasi-analyticity uses complex analysis. Without log-convexity, condition 1 is still equivalent to condition 2 with Mk1/kM_k^{1/k} replaced by inf⁡j≥kMj1/j\inf_{j \ge k} M_j^{1/j}. Condition 3 then has to be applied to the largest log-convex minorant of MM instead of MM.

Example: for Mk=k!M_k = k!, the sequence is log-convex since (k!)2≤(k−1)! (k+1)!(k!)^2 \le (k-1)! \, (k+1)!. The ratios Mk−1/Mk=1/kM_{k-1}/M_k = 1/k have a divergent sum, so the class is quasi-analytic; this is the class of analytic functions on [a,b][a, b]. For Mk=(k!)sM_k = (k!)^s with s>1s > 1, the ratios are k−sk^{-s}, and the sum converges. So these classes contain nonzero bump functions.

Going deeper

For a student

The test is a single series, much like a convergence test for ordinary series. Divergence of ∑Mk−1/Mk\sum M_{k-1}/M_k means that the allowed growth is slow, and the class behaves like the analytic functions. Convergence means that cut-off functions exist in the class, so it supports partitions of unity, as needed in the theory of ultradistributions. The case Mk=(k!)sM_k = (k!)^s gives the Gevrey classes, which appear in the study of partial differential equations.

Reading path

To understand this page from the ground up, read these 18 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 derivativedefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionDenjoy–Carleman theorem, statement, no evidence levelDenjoy–Carleman theor…statement
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. Continuously differentiable function definition
  17. Smooth function definition
  18. Denjoy–Carleman theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quasi-analytic_function, The Denjoy–Carleman theorem
About this page
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Kindstatement, theorem
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MSC26E10
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AboutQuasi-analytic function
UsesQuasi-analytic function, Smooth function, Analytic function, Bump function
Changed2 November 2026

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