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Holomorphic extension of a function whose odd derivatives are summable. (English) Zbl 0581.30024

If I is a real interval, \(f\in C^{\infty}(I)\), and lim inf\(| f^{(2n+1)}(t)| \geq c>0\) for \(t\in I\) then f is the restriction of an entire function. [This could be derived more briefly from the known result that if the radius of convergence of the Taylor series of f at each point is uniformly bounded away from 0 then f is the restriction to I of an analytic function. For a short proof of this result see H. Salzmann and K. Zeller, Math. Z. 62, 354-367 (1955; Zbl 0064.299)].
Reviewer: R.P.Boas

MSC:

30D20 Entire functions of one complex variable (general theory)
26E10 \(C^\infty\)-functions, quasi-analytic functions
26E05 Real-analytic functions

Citations:

Zbl 0064.299
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References:

[1] J. Fischer P. Kolář: On the Validity and Practical Applicability of Derivative Analyticity Relations. J. Math. Phys. 25 (1984), 2538-2544.
[2] I. P. Natanson: Teorija funkcij veščestvennoj peremennoj. Gosud. Izd. Tech.-Teor. Lit. Moskva 1950. · Zbl 0091.05404
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