Estimates of Kolmogorov's \(\varepsilon\)-entropy for compact sets of infinitely differentiable aperiodic functions (Babenko's problem) (Q393839)

Let \(C(I)\) be the space of continuous real functions on the interval \(I=[-1,1]\) with the Chebyshev norm \(\|\cdot\|\) and let \[ G_c^{\infty} := \{f\in C^{\infty}(I)\,: \;\|f\|\leq c, \;\|D^{(p)}f\|\leq G(p) \quad\text{for all}\quad p\in\mathbb{N} \,\}, \] for any fixed constant \(c>0\) and any number sequence \(\{G(p)\}_{p=1}^{\infty}\) satisfying the condition \[ \limsup_{p\to\infty}\sqrt[p]{G(p)}=\infty. \] A compact set \(G_c^{\infty}\) is continuously embedded in the space \(C(I)\) and can be characterized in terms of the expansion coefficients of its elements with respect to the basis in \(C^{\infty}(I)\). Asymptotic estimates of the Kolmogorov \(\varepsilon\)-entropy \(H_{\varepsilon}(G_c^{\infty})\) in terms of the integral of a suitable counting function \(\theta (x)\) are obtained (for the periodic case see [\textit{V. N. Belykh}, Sib. Math. J. 52, No. 3, 381--393 (2011); translation from Sib. Mat. Zh. 52, No. 3, 485-501 (2011; Zbl 1233.46001)]). The research is motivated by concrete needs in computational fluid dynamics.