On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives
For 2π-periodic functions \(x L_∞ ^r \) and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \(|| x^(k) ||_q (v(x^(k))2)^1/q || φ_r-k ||_q||| φ_r |||_p^α ||| x |||_p^α || x^(r) ||_∞^1- α, k, r N, k < r,\) which takes into account the number of changes in the sign of the derivatives ν(x (k)) over the period. Here, α = (r − k + 1/q)/(r + 1/p), ϕ r is the Euler perfect spline of degree r, \(gathered \| | x | \|_p : = sup _a,b R { E_0 (x)_L_p [a,b] :x’(t) e 0 t (a,b)} , E_0 (x)_L_p [a,b] : = inf _c R \| x - c \|_L_p [a,b] , \| x \|_L_p [a,b] : = { _a^b | x(t) |^p dt }^1/p for 0 < p < ∞ , gathered \) and \(\| x \|_L_p [a,b] : = sup vrai _t [ a,b ] | x(t) |\) . The inequality indicated turns into the equality for functions of the form x(t) = aϕ r (nt + b), a, b ∈ R, n ∈ N. We also obtain an analog of this inequality in the case where k = 0 and q = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines., number=4, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Kofanov, V. A., year=2003, month=Apr., pages=456–469