Asymptotics of Kolmogorov's \(\varepsilon \)-entropy for some classes of infinitely differentiable periodic functions (Babenko's problem) (Q606579)

Let \(X\) be a compact set in a Banach space. For \(\varepsilon >0\), let \(H_{\varepsilon}(X) := \log N_{\varepsilon}(X)\) be the Kolmogorov \(\varepsilon\)-entropy of \(X\), where \(N_{\varepsilon}(X)\) is the number of elements in the most economical \((2\varepsilon)\)-covering of \(X\). The problem of searching for \(H_{\varepsilon}(X)\) of a compact set of \(C^{\infty}\)-functions and its asymptotic behavior for \(\varepsilon \to + 0\) was formulated by \textit{K.I.~Babenko} [Basics of Numerical Analysis. RChD, Moscow (2002)]. Let \(C_{2\pi}\) be the Banach space of \(2\pi\)-periodic, continuous functions. Let \(C_{2\pi}^{\infty}\) be the set of all \(2\pi\)-periodic, infinitely differentiable functions. The author considers the compact set \[ X=\{f\in C_{2\pi}^{\infty}; \;\|f\|\leq c,\, \|f^{(p)}\|\leq G(p)\, (p=1,2,\dots)\}\subset C_{2\pi}, \] where \(c>0\) and \(\{G(p)\}_{p=1}^{\infty}\) with \(G(p) > 0\) and \(\limsup_{p\to \infty} G(p)^{1/p} = \infty\) are fixed. An example of such a \(X\) is the Gevrey class. As main result, the author calculates the leading term of \(H_{\varepsilon}(X)\) for \(\varepsilon \to + 0\). The analog problem in the nonperiodic \(C^{\infty}\)-case is still open.