Sharp Remez-Type Inequalities for Differentiable Periodic Functions, Polynomials, and Splines

Abstract

For any $$ \omega $$ > 0, β $$ \epsilon $$ (0, 2 $$ \omega $$ ), and any measurable set B $$ \epsilon $$ I d := [0,d], μB = β, we obtain the following sharp inequality of the Remez type: on the classes S φ (ω) of functions x with minimal period d(d ≥ 2ω) and a given sine-shaped 2 $$ \omega $$ -periodic comparison function '. In particular, we prove sharp Remez-type inequalities on the Sobolev classes of differentiable periodic functions. We also obtain inequalities of the indicated type in the spaces of trigonometric polynomials and polynomial splines.