Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q ∈ (0, ∞], q > p, and s ∈ [p, q], we prove that functions \(x \in L_∞ ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{±} ||_q \leqslant \frac{|| (φ_r)_{±} ||_q}{|| φ_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_∞^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ r is the perfect Euler spline of order r and c s + 1(x) is the constant of the best approximation of the function x in the space L s + 1. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed n, namely, $$|| (τ^(k))_{±} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(·))_{±} ||_q}{|| \cos(·) ||_p} || τ ||_p,$$ where k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated.