Sharp integral inequalities for fractional derivatives of trigonometric polynomials (Q1759364)

The authors study sharp estimates of integral functionals for operators on the set \({\mathcal T}_{n}\) of real trigonometric polynomials \(f_ n\) of degree \(n\geq1\) in terms of the uniform norm \(\| f_ n\|_{C_{2\pi}}\) of the polynomials. In particular, it is shown that the sharp inequality \(\| D^{\alpha}f_ n\|_ q \leq n^{\alpha}\| \cos t\|_{q}\,\| f_ n\|_{\infty}\) holds on the set \({\mathcal T}_{n}\) for the Weyl fractional derivatives \(D^{\alpha}f_ n\) of order \(\alpha \geq 1\) and and also for \(0\leq q<\infty\).