On the tabulation complexity of infinitely differentiable functions and functions specified by the Hölder condition on their derivatives (Q1594266)

Let be given \(n\in\mathbb{N}\), \(0<\alpha\leq 1\) and \(c>0\). The set of all \(n\) times differentiable functions \(f:\mathbb{R}\to [0,1)\) that have the period 1 and satisfy \[ \bigl|f^{(n)} (x)-f^{(n)} (y)\bigr|\leq c|x-y |^\alpha \Gamma(1+r) \] is denoted by \(H_{r,c}\) where \(r=n+\alpha\). The author estimates the \(\varepsilon\)-entropy of \(H_{r,c}\) as well as the 1-entropy of two classes \(H^N_{r,c}\) and \(\widehat H^N_{r,c}\), called the internal discrete analogue and the external discrete analogue of \(H_{r,c}\), respectively, and consisting of functions \(f:\mathbb{Z}\to \{0,1,\dots,N-1\}\).