Landau-type extremal problem for the triple \(\| f\| _{\infty},\| f'\| _{p},\| f''\| _{\infty}\) on a finite interval.
Let \(f\) be a function defined on a given interval \(I\) (finite or infinite). The \(L_p\)-norms are defined by \[ \| f\| _p=\| f\| _{L_p(I)}= \begin{cases} \left(\int\limits_I\left| f\left(x\right)\right| ^p\,dx\right)^{1/p}, & 1\leq p<\infty\\ \mathop{\text{vrai\,sup}}\limits_{x\in I}| f(x)| , & p=\infty, \end{cases} \] as usual. The extremal problem \[ \| f\| _q\leq M_0,\quad \| f^{(n)}\| _r\leq M_n,\quad \| f^{(k)}\| _p\to\sup, \tag{1} \] for any fixed \(k\), \(1\leq k<n\), is known as a problem of Landau-Kolmogorov type. It is well investigated when \(I={\mathbb R}\), \(I={\mathbb R}_+\), or for classes of periodic functions. If the interval \(I\) is finite, the problem is much more difficult. In this case the extremal function depends essentially on the length \(| I| \). Thus, we have an additional parameter that plays an important role. The present article is devoted to the investigation of the problem (1) on a finite interval \(I\) with \(n=2\), \(q=r=\infty\), \(1\leq p<\infty\). In order to formulate the main result, the author defines \[ \Omega^2(I)=\{f\in W^2_\infty(I): \| f\| _\infty\leq 1,\;\| f''\| _\infty\leq 4\}. \] Bounds \(1\) and \(4\) appearing here are chosen according to the Chebyshev polynomial \(T_2(x)=2x^2-1\) and the interval \([-1,1]\). Let \(\Phi\) be the class of all functions \(\varphi\) that are positive, increasing and nonconcave on \((0,\infty)\). Then for every interval \(I\) we define \[ J_\varphi(g;I):= \int_I\varphi (| g(x)|)\,dx. \] Let us note that if \(\varphi(x)=x^p\), the functional \(J_\varphi(g;I)\) reduces to the \(L_p\)-norm of a function \(g\). A parabolic perfect spline with \(m\) knots is a zigzag spline on \([a,b]\) if it has \(m+1\) local extrema on \((a,b)\). The end points of the interval are considered as points of local extremum too. The main result of the article is the following theorem. Theorem. Assume that \(I=[a,b]\) with \(b-a>2\) and \(\varphi\in\Phi\). Then every extremal function of the problem \[ J_\varphi(f';I)\to\sup,\quad \text{over all }f\in\Omega^2(I), \tag{2} \] is a zigzag perfect parabolic spline on \([a,b]\). Moreover, all its extrema, except eventual one, equal \(1\) in theabsolute value. The case \(| I| \leq 2\) is considered in [\textit{B. Bojanov} and \textit{N. Naidenov} [J. Approximation Theory 117, 55--73 (2002; Zbl 1019.41009)]. Some properties of parabolic splines are investigated in the article. The class of extremal functions is discussed in the last section. At the end of the article the author poses two questions that seem to be very interesting: 1. Does the analog of the mentioned theorem hold for \(0<p<1\)? 2. Can problem~(2) be solved in a constructive way? The article is highly recommended for researchers in approximation theory.
scientific article; zbMATH DE number 1970534