Szegö inequality for trigonometric polynomials in \(L_p , 0 \leq p \leq \infty ,\) with the classical value of the best constant (Q2077258)

For trigonometric polynomials \[ f_n(t)=\frac{a_0}{2}+\sum_{k=1}^{n}\bigl(a_k\cos(kt)+b_k\sin(kt)\bigr) \] the Weyl derivative (fractional derivative) \(D^{\alpha}f_n\) for \(\alpha\geq0\) is defined by \[ D^{\alpha}f_n(t)=\sum_{k=1}^{n}k^{\alpha}\bigl(a_k\cos(kt+\tfrac{\pi\alpha}{2}) +b_k\sin(kt+\tfrac{\pi\alpha}{2})\bigr). \] For \(\alpha\in{\mathbb N}\), it coincides with the classical derivative. Moreover, for \(\theta\in{\mathbb R}\), the author considers the so-called Weyl-Szegö operator \[ D_{\theta}^{\alpha}f_n(t)=\sum_{k=1}^{n}k^{\alpha}\bigl(a_k\cos(kt+\tfrac{\pi\alpha}{2}+\theta) +b_k\sin(kt+\tfrac{\pi\alpha}{2}+\theta)\bigr). \] The main result of the paper is the inequality \[ {\|D_{\theta}^{\alpha}f_n\|}_{p}\leq n^{\alpha}{\|f_n\|}_{p}, \] which holds for all \(n\in{\mathbb N}\), \(\alpha\geq2n-2\), \(\theta\in{\mathbb R}\), and \(0\leq{p}\leq\infty\). The factor \(n^{\alpha}\) is thereby best possible. This complements results of Arestov and Glazyrina.