# Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

• V. A. Kofanov

### Abstract

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 $x \in L_\infty ^{{\text{ }}r}$ satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ 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, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where kN, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated.
Published
25.05.2001
How to Cite
Kofanov, V. A. “Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 53, no. 5, May 2001, pp. 597-09, https://umj.imath.kiev.ua/index.php/umj/article/view/4283.
Issue
Section
Research articles