The Jackson interpolation operator and construction of functions
For \(x_{\mu }=2\mu \pi /n,\) \(\mu =0,\pm 1,\pm 2,\dots\) and \(f\in C_{2\pi },\) let \[ J_{n}(f;x)=\frac{1}{n^{2}}\sum_{\mu =0}^{n-1}f(x_{\mu })\left( \frac{\sin \frac{n}{2}(x_{\mu }-x)}{\sin \frac{1}{2}(x_{\mu }-x)}\right) ^{2} \] be the Jackson interpolation operator of degree \(n.\) The authors obtain the equivalence of the error approximation \(\| J_{n}(f)-f\| \) to the \(K\)-functional \(K(f,t)=\inf \{\| f-g\| +t\| g^{\prime }\| +t\| \widehat{g}^{\prime }\| :g\in C_{2\pi }^{2}\}.\) More exactly, the following relation is established: \[ \tfrac{1}{n} \max_{1\leq k\leq n}\| J_{k}(f)-f\| \approx K(f,\tfrac{1}{n}),\quad \forall f\in C_{2\pi } \] where the symbol ``\(\approx\)'' does not depend on \(n\) and \(F.\) Also, if \(f\) and its conjugate \(\widehat{f}\) are in \(C_{2\pi },\) one obtains the following asymptotic formula \[ K(f,\delta )+K(\widehat{f},\delta )\approx \omega (f,\delta )+\omega (\widehat{f},\delta ), \] where the symbol ``\(\approx\)'' does not depend of \(f\), \(\widehat{f}\) and \(\delta .\) Finally if \(\omega (\delta )\) is a modulus of continuity such that \[ \int_{0}^{\delta }\frac{\omega (t)}{t}\,dt=O(\omega (\delta )), \] then for \(f\in C_{2\pi }\) one has \[ \| J(f)-f\| =O(\omega (\tfrac{1}{n} ))\iff \omega (f,\delta )+\omega (\widehat{f},\delta )=O(\omega (\delta )). \]
scientific article; zbMATH DE number 5082287