Sharp Kolmogorov–Remez-Type Inequalities for Periodic Functions of Low Smoothness

• Published in: Ukrainian mathematical journal 2020-09, Vol.72 (4), p.555-567
• Main Author: Kofanov, V. A.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-020-01800-2

In the case where either r = 2, k = 1 or r = 3, k = 1, 2, for any q , p ≥ 1, β ∈ [0, 2π), and a Lebesgue-measurable set B ⊂ I 2π ≔ [−π/2, 3π/2], μB ≤ β , we prove a sharp Kolmogorov–Remez-type inequality f k q ≤ φr − k q E 0 φr L α p I 2 π / B 2 m f L p α I 2 π / B F r ∞ 1 − α , f ∈ L ∞ r , with α = min {1 − k / r , ( r − k + 1/ q )/( r + 1/ p )}, where φ r is the perfect Euler spline of order r, E 0 f L p G is the best approximation of f by constants in Lp ( G ) , B 2 m = π − 2 m 2 π + 2 m 2 , and m = m ( β ) ε [0 , π) is uniquely defined by β. We also establish a sharp Kolmogorov–Remez-type inequality, which takes into account the number of sign changes of the derivatives.