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Lower bounds for quasianalytic functions. II: The Bernstein quasianalytic functions. (English) Zbl 1064.30033
In this paper the authors consider questions related to Bernstein quasianalytic classes that are similar to those raised in the paper by F. Nazarov, M. Sodin and A. Volberg [Math. Scand. 95, 59–79 (2004; Zbl 1064.30032)] for smooth Denjoy-Carleman quasianalytic functions. A function $$f$$ continuous on $$I:=[-1,1]$$ is said to be quasianalytic in the sense of S.N. Bernstein, if for some $$\beta>0$$ $E_n(f):=\min_{P\in \mathcal P_n}\| f-P\| _I\leq e^{-\beta n}\tag{$$\spadesuit$$}$ when $$n$$ runs through an infinite subsequence $$\{n_j\}\subset\mathbb N$$. Here $$\mathcal P_n$$ is the space of all algebraic polynomials of degree $$\leq n$$. Let $$B(I)$$ denote the set of all such functions. A classical result of Bernstein states that if $$f\in B(I)$$ vanishes on a subset of I of positive measure, then $$f$$ is the zero function. Actually, it is sufficient to assume above that $$f$$ vanishes on a subset of $$I$$ of positive logarithmic capacity [see H. Szmuszkowiczówna, C. R. Acad. Sci., Paris 198, 1119–1120 (1934; Zbl 0008.36501); P. Lelong, C. R. Acad. Sci., Paris 224, 883–885 (1947; Zbl 0031.05104)]. If the sequence $$\{n_j\}$$ of $$(\spadesuit)$$ is not too lacunary: $$\limsup_{j\to\infty}\frac{n_{j+1}}{n_j}<\infty$$, then $$f$$ is a germ of an analytic function on $$I$$. Having the uniqueness property, generally speaking, the functions of the class $$B(I)$$ do not posses any smoothness. It is a result of S. Mazurkiewicz [Mathematica, Cluj 13, 16–21 (1937; Zbl 0018.13403)] that the subset of $$B(I)$$ formed by nowhere differentiable functions is residual in the space $$C(I)$$ of all continuous functions on $$I$$ (i.e. $$C(I)\setminus B(I)$$ is of the first (Baire) category).
The main result of this paper is the following asymptopic upper bound for the size of the level sets $m_f(t)=| \{x\in[I: | f(x)| \leq t\}|$ for $$t=E^*_{n_j}(f):=\max(E_n(f),e^{-n})$$:
Theorem A. Suppose $$f$$ satisfies $$(\spadesuit)$$ with sufficiently lacunary sequence $$\{n_j\}: \lim_{j\to\infty}\frac{n_{j+1}}{n_j}=\infty.$$ Then $\lim_{j\to\infty}\frac{| \log m_f(E^*_{n_j+1}(f))| }{| \log m_f(E^*_{n_j}(f))| }= +\infty.$ The second result of the paper shows that the Bernstein quasianalytic functions may have deep zeros of prescribed flatness:
Theorem B. Given decreasing functions $$\phi,\psi: [1,+\infty)\to (0,+\infty)$$, $$\lim_{t\to\infty}\phi(t)= \lim_{t\to\infty}\psi(t)=0$$, there exist a function $$f\in C(I)$$ and a subsequence $$\{n_j\}\subset\mathbb N$$ such that for $$n\in\{n_j\}$$ $E_n(f)\leq\psi(n),\tag{$$†$$}$ and $| f(x)| \leq e^{-n},\quad | x| \leq\phi(n).$ In particular, if $$\psi(s)=e^{-s}$$, one gets $m_f(E^*_{n_j}(f))\geq 2\phi(n_j),\quad j=1.2.\dots .$
Reviewer’s remark. Let us note that the existence of a function $$f\in C(I)$$ satisfying $$(†)$$ immediately follows from the well-known Bernstein lethargy theorem.

##### MSC:
 30D60 Quasi-analytic and other classes of functions of one complex variable
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