Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms

摘要

We consider the Bernstein-Jackson-Nikol'skii inequalities for fractional derivatives in the case when the norm is asymmetric. Assume that n ∈ N, p1,p2,q1,q2 ∈ [1, ∞], and α ∈ R_. Then sup (||D~αt_n||_(q1,q2))/(||t_n||_(p1,p2)) = I_αn~(α+ψ_1(p1,p2,q1,q2)) + n~(α+ψ_2(p1,p2,q1,q2)), t_n ∈ τ_n t_n not≡ 0 where I_α = {α, 0 ≤ α ≤ 1, 1, α ≥ 1, and the functions ψ_1 and ψ_2 are given by an explicit formula. The asymptotic behaviour is with respect to n for fixed α p1,p2,q1, and q2.