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

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 k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated.