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Lower bounds for quasianalytic functions. I: How to control smooth functions. (English) Zbl 1064.30032
This paper is a variation on Bang’s doctoral thesis concerning Denjoy-Carleman classes of \(C^\infty\)-functions, published 50 years ago. The authors claim that Bang’s thesis left unsufficient trace in the literature devoted to quasianalytic functions. Therefore they took liberty to reproduce some Bang’s results with their proofs. Given a non-decreasing function \(A:\;[1,\infty)\to(0,\infty)\), set \(M_0=1\) and \(M_j=M_{j-1}A(j)\), \(j\geq 1\) and define the (normalized) Denjoy-Carleman class \(\mathcal C_A([0,1])\) to be the set of \(C^\infty([0,1])\)-functions such that \(\| f^{(j)}\| _{[0,1]}\leq M_j\), \(j\in\mathbb Z_+\). By the classical Denjoy-Carleman theorem, the class \(\mathcal C_A([0,1])\) is quasianalytic (i.e. it contains no non-trivial function that vanishes at a point with all derivatives) if and only if \(\sum_{j=1}^{\infty}\frac{M_{j-1}}{MM_j}=\infty\). The Bang degree \(\mathfrak n_f\) of \(f\in\mathcal C_A([0,1])\) is the largest integer \(N\) such that \(\sum_{\log\| f\| ^{-1}_{[0,1]}<j\leq N}\frac{M_{j-1}}{M_j}<e.\) The following theorems show that Bang’s degree is an important characteristics of smooth functions.
Theorem A (Bang 1953). The total number of zeroes (counting with multiplicities) of \(f\in\mathcal C_A([0,1])\) does not exceed its Bang degree.
Assume moreover that the function \(A\) is a \(C^1\)-function. Set \[ \gamma(n):=\sup_{1\leq s\leq n}\frac{sA^\prime(s)}{A(s)}\quad\text{and}\quad \Gamma(n)=4e^{4+\gamma(n)}. \]
Theorem B. Suppose \(f\in\mathcal C_A([0,1])\). Then for any interval \(I\subset[0,1]\) and any measurable subset \(E\subset I\) \[ \| f\| _I\leq\left(\frac{\Gamma(2\mathfrak n_f)| I| }{| E| }\right)^{2\mathfrak n_f}\| f\| E. \]
The above inequality is in the spirit of the classical Remez and Bernstein inequalities for polynomials. Therefore it seems to be natural to ask how to band from above the Bang degree of a polynomial by its usual degree. The same question can be asked about the upper bound of the Bang degree of a real analytic function \(f\) by its Bernstein degree \(\mathfrak B_f(K,G):=\log(\| f\| _G/\| f\| _K)\). The authors leave these questions open.

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