The Bernstein-Szegö inequality for fractional derivatives of trigonometric polynomials

Abstract

On the set F _n of trigonometric polynomials of degree n ≥ 1 with complex coefficients, we consider the Szegö operator $D_\theta ^\alpha$ defined by the relation $D_\theta ^\alpha f_n (t) = \cos \theta D^\alpha f_n (t) - \sin \theta D^\alpha \tilde f_n (t)$ for α, θ ∈ ℝ, where α ≥ 0. Here, $D^\alpha f_n$ and $D^\alpha \tilde f_n$ are the Weyl fractional derivatives of (real) order α of the polynomial f _n and of its conjugate $\tilde f_n$. In particular, we prove that, if α ≥ n ln 2n, then, for any θ ∈ ℝ, the sharp inequality $\left\| {\cos \theta D^\alpha f_n - \sin \theta D^\alpha f_n } \right\|_{L_p } \leqslant n^\alpha \left\| {f_n } \right\|_{L_p }$ holds on the set F _n in the spaces L _p for all p ≥ 0. For classical derivatives (of integer order α ≥ 1), this inequality was obtained by Szegö in the uniform norm (p = ∞) in 1928 and by Zygmund for 1 ≤ p < ∞ in 1931–1935. For fractional derivatives of (real) order α ≥ 1 and 1 ≤ p ≤ ∞, the inequality was proved by Kozko in 1998.