Jackson's theorems in \(L_ 2\) (Q1099343)

The paper deals with a generalization of the classical Jackson inequalities \(E_{n-1}(f)\ll n^{-r}\sup_{| t| \leq 1/n}\| \Delta^ m_ tf^{(r)}\|;\) m,n\(\in {\mathbb{N}}\), \(r=0,1,2,..\). for \(f\in L_ 2(-\pi,\pi)\) where \(\|.\|\) is the norm in \(L_ 2(- \pi,\pi)\). In this paper the inequality \[ E_{n-1}(f)\ll n^{-r}\max \{\| \Delta^ m_{2\pi \alpha_ 1/n}f^{(r)}\|,\quad \| \Delta^ m_{2\pi \alpha_ 2/n}f^{(r)}\| \] is proved for all \(r\geq m\), where \(\alpha_ 1\), \(\alpha_ 2\), are such numbers that \(0<\alpha_ 1<\alpha_ 2<1\) and that \(\alpha_ 1/\alpha_ 2\) is a special number satisfying some arithmetic conditions.