2017
Том 69
№ 9

All Issues

Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

Kofanov V. A.

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Abstract

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions xL x (r), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ where $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$ k, rN, 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.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 10, pp 1627-1636.

Citation Example: Kofanov V. A. Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications // Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1348-1356.

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