Approximation of smooth functions on the semiaxis by entire functions of bounded half-degree (Q5961086)

Let \(W^r\) (\(r\) a natural number) be the class of functions \(f\) on the positive real semiaxis \([0, +\infty)\) whose derivative \(f^{(r-1)}\) is absolutely continuous and \(|f^{(r)}|\leq 1\) holds almost everywhere, and let \(H_\sigma\), \(\sigma>0\), be the class of entire functions of bounded half-degree of type at most \(\sigma\). The author studies pointwise approximation of functions \(f\in W^r\) by functions in \(H_\sigma\). The main result of the paper: Let \(f\in W^r\). Then for any \(\sigma>0\) there exists a function \( g\in H_\sigma\) for which \[ |f(x)-g(x)|\leq 2^r{K_r\over \sigma^r}x^{r/2} + {\gamma \over \sigma^{r+1}} x^{(r-1)/2} \] holds for all \(x\geq 0\) with a positive constant \(\gamma\) depending only on the natural number \(r\). The best possible value of the constant \(K_r\) is given by \(K_r=4/\pi\sum_{k=0}^\infty (-1)^{k(r+1)}/(2k+1)^{(r+1)}\).