The oscillation of derivatives: the Bernstein problem for Fourier integrals
A classical theorem due to S. Bernstein asserts that if a real-valued function f and all its derivatives are of constant sign in a closed interval I, then f is analytic in I. \textit{G. Szegö} [Trans. Am. Math. Soc. 52, 450-462 (1942; Zbl 0060.195)] sharpening an earlier result of \textit{G. Polya} and \textit{N. Wiener} [ibid. 52, 249-256 (1942; Zbl 0060.195)] proved the following results: Let \(f\in C^{\infty}_{2\pi}\) and let \(N_ k\) denote the number of distinct sign changes of \(f^{(k)}\) within a period. Then (1) \(N_ k=O(k/\log k)\) implies that f is an entire function; (2) \(N_ k=O((k/\rho)^{\rho})\) \((\rho >1)\) implies that f is an entire function of finite order \(\leq \rho /(\rho -1)\) and (3) \(N_ k=0(1)\) implies that f is a trigonometrical polynomial. These results are essentially best possible. In this paper the author proves similar although not so sharp results for \(C^{\infty}\) real-valued functions representable as Fourier transforms.
scientific article; zbMATH DE number 3946845