The generalized Marcinkiewicz-Zygmund inequality for trigonometric polynomials
Let \(T\) be a given trigonometric polynomial of degree \(n\) and \(p\) be a number such that \(1<p<+\infty\). The well-known Marcinkiewicz-Zygmund inequality [see \textit{A. Zygmund}: ``Trigonometric series'', Vol. II (1959; Zbl 0085.056), p. 28] provides an estimate for \((*) [\int_ 0^{2\pi}| T(x)|^ pdx]^{1/p}\) in terms of a sum involving the values of \(T\) at certain equispaced points in \([0,2\pi]\). The author generalises this inequality to trigonometric polynomials of a certain class in such a way that \((*)\) is estimated in terms of the values of the polynomial and its derivatives at prescribed equispaced points. Such a generalisation is then used to prove the mean convergence of the Hermite and some other interpolants.
scientific article; zbMATH DE number 57724
scientific article