An inverse theorem of approximation theory of periodic functions in various metrics (Q1206258)

In connection with a problem of S. B. Stechkin the author solves the following problem: given a rate \(E_{n-1}(f)\leq\varepsilon_ n\) with some monotone and fixed null-sequence \(\{\varepsilon_ n\}\) for the best trigonometric approximation of a function \(f\), determine the largest possible value of the \(k\)-th modulus of smoothness \(\omega_ k\left(f^{(r)};{\pi\over n}\right)\) of \(f^{(r)}\) in \(L_ \infty\) norm (the determination in some \(L_ q\) norm was done before by the author). It turns out that the order of the supremum is \[ \sum^ \infty_{\nu=n+1}\nu^{\sigma-1}e_ \nu+n^{-k}\sum^ n_{\nu=1}\nu^{k+\sigma-1} \varepsilon_ n, \] where \(\sigma=r_ 1/p\).