On inequalities of Bohr and Bernstein (Q701400)

The author proves inequalities of Bohr-Favard and Bernstein for \(N_\Phi\)-spaces generated by concave functions \(\Phi\). Among others he proves: Let \(f\) and its generalized derivative \(f^{(n)}\) be in \(N_\Phi(\mathbb{R})\) and \(\sup\widehat f\cap(- \sigma,\sigma)= \emptyset\). Then \(f^{(k)}\in N_\Phi(\mathbb{R})\) for all \(0< k< n\) and \[ \|f\|_{N_\Phi}\leq \sigma^{- n}K_n\|f^{(n)}\|_{N_\Phi}, \] where \(\widehat f\) is the Fourier transform of \(f\) and \(K_n\) is the \(n\)th Favard constant. An analogous theorem is proved on the torus, and some further pleasant results can be found in the paper.