Sharp Remez-type inequalities of various metrics in the classes of functions with а given comparison function - Gaydabura, A. E.; Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
For any p [1,∞ ], ω > 0, β (0, 2ω ), and any measurable set B I_d := [0, d], μ B ≤ β, we obtain the following sharp Remez-type inequality of various metrics E_0(x)∞ ≤ \| φ \|_∞ E_0 (φ )L_p(I_2ω B_1)\| x\|_ L_p(I_dB) on the classes S_φ (ω ) of d-periodic (d ≥ 2ω) functions x with a given sine-shaped 2ω -periodic comparison function φ, where B_1 := [(ω β )/2, (ω + β )/2], E_0(f)L_p(G) is the best approximation of the function f by constants in the metric of the space L_p(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.