Examples of Landau--Kolmogorov inequality in integral norms on a finite interval
Given a non-negative integer \(m\) and a compact interval \([a,b]\), let \(T_{r,m}\) be the unique (Chebychev) perfect spline of degree \(r\) with \(m\) knots in \((a,b)\) satisfying the conditions \(\|T_{r,m}\|_{C[a,b]}=1\) and \(T_{r,m}(t_j)=(-1)^{m+r-j},\;j=0,\dots,m+r\), for some points \(a=t_0<\dots<t_m=b\). In the paper the following Landau-Kolmogorov-type inequality for \(r=\)2 and 3 is proved: for each function \(f\in W^r_\infty([a,b])\) such that \(\|f\|\leq 1\) and \(\|f^{(r)}\|\leq \|T^{(r)}_{r,m}\|\) one has \[ J_{\varphi}(f^{(k)})\leq J_{\varphi}(T^{(k)}_{r,m})\quad \text{for} 1\leq k\leq r-1. \] Here \(J_{\varphi}(h)\) stands for the norm \(\int_a^b\varphi(|h(x)|)dx\) with an \(N\)-function \(\varphi\) (e.g. \(\varphi(x)=x^p\), \(1\leq p<\infty\)). An interesting historical introduction to the problem is also made by the authors.
scientific article; zbMATH DE number 1891483