On inequalities of the Landau-Kolmogorov-Hörmander type on a segment and the real line (Q2730511)

Let \(x(t)\) be a real-valued function on \([-1,1 ]\) such that \(x^{(r)}\in L_\infty\), let \(E_0(x)_\infty\) be the best uniform approximation of \(x\) by a constant, and let \(x_\pm(t)=\max\{\pm x(t),0\}\). Suppose that for some given constants \(A,\alpha ,\beta >0\) NEWLINE\[NEWLINE E_0(x)_\infty \leq A, \quad \|x^{(r)}_+\|_{[-1,1 ]}\leq \alpha , \quad \|x^{(r)}_-\|_{[-1,1 ]}\leq \beta NEWLINE\]NEWLINE (\(L_\infty\)-norms). The author gives an exact estimate for \(\|x^{(k)}\|_{[-a,a ]}\) on a subinterval \([-a,a ]\subset [-1,1 ]\), \(k=1,2,\ldots ,r-1\). The method is based on spline approximations.