Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications
We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x ∈ L ∞ x (r), namely, \| x^(k) \|_L_∞ (R) \| φ _r - k \|_∞ \| φ _r \|_∞ ^1 - k/r M(x)^1 - k/r \| x^(r) \|_L_∞ (R)^k/r , where M(x): = 12_α ,β { | x(β ) - x(α ) |:x’(t) e 0 t (α ,β ) }, k, r ∈ N, k < r, and ϕ r is a perfect Euler spline of order r. Using this inequality, we strengthen the Bernstein inequality for trigonometric polynomials and the Tikhomirov inequality for splines. Some other applications of this inequality are also given., number=10, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Kofanov, V. A., year=2002, month=Oct., pages=1348–1356