Degree of approximation of functions in the Hölder metric by Borel's means (Q5916368)

Let \(C_{2\pi}\) denote the space of all \(2\pi\)-periodic, continuous functions. For \(0<\alpha\leq 1\) and some positive constant \(K\) the Banach space \(H_ \alpha\) is given by \[ H_ \alpha=\{f\in C_{2\pi}:| f(x)-f(y)|\leq K| x-y|^ \alpha\} \] with the norm \[ \| f\|_ \alpha=\| f\|_ C+\sup\{\Delta^ \alpha f(x,y)\}, \] where \(\| f\|_ C=\sup_{-\pi\leq x\leq\pi}| f(x)|\) and \(\Delta^ \alpha f(x,y)=| x-y|^{-\alpha}| f(x)-f(y)|\;(x\neq y),\Delta^ 0f(x,y)=0.\) Let \(s_ n(f;x)\) be the \(n\)-th partial sum of the trigonometric Fourier series. \[ B_ p(f;x)=\exp(-p)\sum^ \infty_{n=0}s_ n(f;x)p^ n/n!\quad(p>0) \] denotes the Borel exponential mean of \((s_ n(f;x))\). In this paper the following two theorems are proved. Theorem 1. Let \(0\leq\beta<\alpha\leq 1\) and let \(f\in H_ \alpha\). Then \(\| B_ p(f)-f\|_ \beta=O\{p^{\beta-\alpha}\cdot\log p\}\). Theorem 2. Let \(f\in H_ \alpha(0<\alpha\leq 1)\). Then, if \[ \int^{\log p/\sqrt p}_{\pi/p}t^{-1}|\varphi_ x(t+\pi/p)-\varphi_ x(t)|\exp(-pt^ 2/3)dt=O (p^{-\alpha})\;(2\varphi_ x(t)=f(x+t)+f(x-t)-2f(x)), \] \[ \| B_ p(f)-f\|_ \beta=O\{p^{\beta-\alpha}(\log p)^{\beta/\alpha}\} (0\leq\beta<\alpha\leq 1). \]