V.G. Timofeev. N.P. Kuptsov’s method for the construction of an extremal function in an inequality between uniform norms of derivatives of functions on the half-line ... P. 220-239

On the class $L_\infty^4(\mathbb{R}_+)$ of functions $f\in C(\mathbb{R}_+)$ having a locally absolutely continuous third-order derivative on the half-line $\mathbb{R}_+$ and such that $f^{(4)}\in L_\infty(\mathbb{R}_+)$, we study an extremal function in the exact inequalities
$$
\| f^{(j)} \| \leq C_{4,j}(\mathbb{R}_+)\, \| f\|^{1-j/4} \, \| f^{(4)} \|^{j/4},\quad j=\overline{1,3},\quad f\in L_\infty^4(\mathbb{R}_+).
$$
We present N.P. Kuptsov's earlier unpublished method for the construction of an extremal function, which is an ideal spline of the fourth degree. The method is iterative; it finds the knots and coefficients of the spline and calculates the values $C_{4,j}(\mathbb{R}_+)$. The proposed approach differs from the approach of Schoenberg and Cavaretta (1970) and allows to understand the structure of the problem more deeply.

Keywords: inequality between norms of derivatives of functions, four times differentiable functions, uniform norm, half-line

Received December 9, 2018

Revised May 6, 2019

Accepted May 20, 2019

Vladimir Grigor’evich Timofeev, Cand. Sci. (Phys.-Math.), Saratov State University, Saratov, 410012 Russia, e-mail: timofeevvg48@gmail.com

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Cite this article as: V.G.Timofeev. N.P.Kuptsov’s method for the construction of an extremal function in an inequality between uniform norms of derivatives of functions on the half-line, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 220–239.