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Estimates of the derivatives of polynomials on convex bodies. (English. Russian original) Zbl 0921.46034
Lupanov, O. B. (ed.), Analytic number theory and applications. Collected papers in honor of the sixtieth birthday of Professor Anatolii Alexeevich Karatsuba. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 218, 372-383 (1997); translation from Tr. Mat. Inst. Steklova 218, 374-384 (1997).
Let \(X\) and \(Y\) be a real Banach spaces. By \({\mathfrak P}_n (X,Y)\) we denote the space of all continuous polynomials from \(X\) into \(Y\) of degree not higher than \(n\). For a convex bounded closed body \(V\) in \(X\) define \(\| P_n\|_V= \sup_{x\in V}\| P_n(x)\|_Y\) and \(\| P_n'\|_V= \sup_{x\in V}\| P_n'(x)\|_{L(X,Y)}\). The width \(\omega_V\) of the body \(V\) is defined as follows: For \(h\in S_X\) define \[ \begin{aligned} V(h)&= \{(y_1,y_2): y_1,y_2\in V\text{ and }\exists c\in\mathbb{R}: y_1- y_2= c\cdot h\},\\ \omega_V(h)&= \sup_{(y_1,y_2)\in V(h)}\| y_1-y_2\|, \text{ and }\omega_V= \inf_{\| h\|=1} \omega_V(h). \end{aligned} \] Finally, set \(r(V):= \sup\{r: \exists x_0\in V: B(x_0,r) \subseteq V\}\). A. V. Andrianov proved that \[ \| P_n'\|_V\subseteq \frac{4n^2}{r(V)} \| P_n\|_V\text{ holds true for all }P_n\in{\mathfrak P}_n(X,Y), \] [see A. V. Andrianov, Mat. Zametki 52, No. 5, 13-21 (1992; Zbl 0845.46021)]. In [V. I. Skalyga, Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 1, 141-156 (1997; Zbl 0889.41008)] the author proved the stronger estimate \[ \| P_n'\|_V\leq \frac{4n^2}{\omega_V} \| P_n\|_V. \] In this paper the author combines several techniques to obtain the unimprovable estimate \[ \| P_n'\|_V\leq 2n\text{ cot} \biggl( \frac{\pi}{4n} \biggr) \frac{\| P_n\|_V}{\omega_V}. \]
For the entire collection see [Zbl 0907.00013].

46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
46G20 Infinite-dimensional holomorphy