Comparison of the orders of approximation of the function classes \(W^ RH_{\omega}\)
Let \(E_ n({\mathcal J})\) be the best uniform approximation of the class F of functions by trigonometric polynomials of order not exceeding n; \((*)\quad {\mathcal E}(W^ rH_{\omega};{\mathcal L}_ n)=\sup_{f\in W^ rH_{\omega}}\| f(x)-{\mathcal L}_ n(f;x)\| C,\quad ({\mathcal L}_ n(f;x)=(1/\pi)\int^{\pi}_{- \pi}f(x+t)\{+\sum^{n}_{k=1}\lambda_{k,n}\cos kt\}dt,\quad n=1,2,...).\) The main result can be stated as follows: If \(0\leq r_ 1<r_ 2\), \(\omega_ 1(t)\) and \(\omega_ 2(t)\) are any moduli of continuity, then for any linear method of summation of Fourier series (*) one has \[ {\mathcal E}(W^{r_ 1}H_{\omega};{\mathcal L}_ n)\leq C(n^{r_ 2-r_ 1}/\omega_ 2(1/n))\omega_ 1(1/n){\mathcal E}(W^{r_ 2}H_{\omega_ 2};{\mathcal L}_ n). \]
scientific article; zbMATH DE number 3929637