Sharp Remez-type Inequalities of Various Metrics in the Classes of Functions with Given Comparison Function
For any p ϵ [1,∞], 𝜔 > 0, β ϵ (0, 2𝜔), and any measurable set B ⊂ I d := [0,d], μB ≤ β, we establish a sharp Remez-type inequality of various metrics
in the classes S𝜑(ω) of d-periodic (d ≥ 2ω) functions x with a given sine-shaped 2ω-periodic comparison function 𝜑, where B1 := [(ω − β)/2, (ω+β)/2] and E0(f)Lp(G) is the best approximation of the function f by constants in the metric of the space Lp(G). In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.
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