Bounds of the Nikol'skii polynomial constants in \(L^p\) with Gegenbauer weight
The authors estimate a sharp Nikol'skii constant in the inequality \(\|u\|_{\infty}\leq \mathcal{C}_{\alpha}(n)\|u\|_p\) for trigonometric and algebraic polynomials of degree at most \(n\) in the space \(L^p\) on \((-\pi,\pi]\) with the periodic Gegenbauer weight \(|\sin x|^{2\alpha +1}\) and on \([-1, 1]\) with the algebraic Gegenbauer weight \((1-x^2)^{\alpha}\), respectively. Suppose that \(p\geq 1\) and let \(\mathcal{L}_p\) be a sharp Nikol'skii constant for entire functions of exponential type at most 1 in the space \(L^p\) on \(\mathbb{R}\) with the power weight \(|x|^{2\alpha +1}\). It is proved that, for all \(\alpha\geq -1/2\), \[ n^{(2\alpha + 2)/p}\mathcal{L}_p\leq\mathcal{C}_{\alpha}(n)\leq (n + 2s_{p,\alpha})^{(2\alpha + 2)/p}\mathcal{L}_p, \] where \(s_{p,\alpha}\) does not depend on \(n\). These bounds make it possible to refine some known estimates of the Nikol'skii constants.
scientific article; zbMATH DE number 7466407
scientific article