2017
Том 69
№ 9

All Issues

On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives

Kofanov V. A., Miropol'skii V. E.

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Abstract

New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period.

English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 12, pp 1927-1936.

Citation Example: Kofanov V. A., Miropol'skii V. E. On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives // Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1642–1649.

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