Boyanov–Naidenov problem and the Kolmogorov-type inequalities for functions on the real axis - Kofanov, V. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.5 We solve the Boyanov–Naidenov problem \|x^(k)\|_q, δ → , k= 1,… ,r-1, q 1, on the classes of functions W^r_p,ε(A_0, A_r):={xL^r_∞\|x\|_p, ε A_0 , \|x^(r)\|_∞ A_r }, where \|x\|_p, δ:={ \|x\|_L_p[a, b]a, b R, 0< b-a δ }, p, δ > 0, ε (0, ε_1 ], ε_1 := π / ω, the number ω satisfies the condition A_0 = A_r \|φ_ω, r\|_p, π / ω, φ_ω, r(t):=ω^-rφ_ r(ω t), and φ_ r is the ideal Euler spline of order r. In addition, we prove that the Boyanov–Naidenov problem is equivalent to the problem of sharp constant C = C(λ) in the Kolmogorov-type inequality gather\|x^(k)\|_q, δ ≤ C \|x\|_p, ε^α \|x^(r) \|_∞^1-α, xL^r, λ_p,ε, 1 gather where α=r-k+1/qr+1/p, L^r, λ_p,ε:= {xL^r_∞\|x\|_p, ε = \|φ_λ, r\|_p, ε· \|x^(r)\|_∞ }, and λ > 0. In particular, we obtain the sharp inequality of the form (1). We also solve the Boyanov–Naidenov problem in the spaces of trigonometric polynomials and splines and establish the theorems on the relationship between this problem and sharp inequalities of the Bernstein type. As a consequence, we prove sharp inequalities of the indicated type for polynomials and splines.