Widths in \(L_ 2\) of classes of differentiable functions, defined by higher-order moduli of continuity
Using the standard notations, the author proves the following theorem: If \(f^{(r)}\in L_ 2\) and \(f(x)\sim (a_ 0/2)+\sum^{\infty}_{k=1}\rho_ k \cos (kx+\phi_ k),\) then for any \(n,k=1,2,...\), \(r=0,1,..\). \[ (*)\quad \sup_{f^{(r)}\in L_ 2;f\neq const}n^{r-k/2} E_ n(f)\{\int^{\pi /n}_{0}\omega_ k^{2/k}(f^{(r)};t)\sin nt dt\}^{-k/2}=2^{-k}, \] where \(E_ n(f)=(\pi \sum^{\infty}_{k=n}\rho^ 2_ k)^{1/2}.\) He also gives a generalization of (*) for the class \(W^ r_ k\psi (\subset L^ r_ 2)\).
scientific article; zbMATH DE number 4208759