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

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We present N. P. Kuptsov's earlier unpublished method for the construction of an extremal function, which is an ideal spline of the fourth degree.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.