Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines
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 \(x L_∞ ^ r\) satisfy the unimprovable inequality || (x-c_s+1 (x))_± ||_q || (φ_r)_± ||_q|| φ_r ||_p^r+1/qr+1/p || x-c_s+1(x) ||_p^r+1/qr+1/P || x^(r) ||_∞^1/p-1/qr+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 n^k+1/p-1/q || ((·))_± ||_q|| (·) ||_p || τ ||_p, where k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated., number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Kofanov, V. A., year=2001, month=May, pages=597–609