Sharp inequalities for the norms of conjugate functions and their applications (Q1099354)

Let C and \(L_ p\), \(1\leq p\leq \infty\) be the spaces of all \(2\pi\)- periodic functions \(f: {\mathbb{R}}\to {\mathbb{R}}\) with the norms \(\| \cdot \|_ c\), \(\| \cdot \|_{L_ p}=\| \cdot \|_ p\) correspondingly and \(L^ r_ p=\{f\in C:f^{(r)}\) exists and belongs to \(L_ p\}\). The author obtains the exact inequalities of Kolmogorov type estimating \(\| \tilde f^{(k)}\|\) by \(\| f\|\) and \(\| f^{(r)}\|\) (\(\sim\) means conjugation) for f from \(L^ r_ p\) or some more general spaces. The analogous results for classes of \(2\pi\)-periodic polynomial splines are also obtained.