The Marcinkiewicz-Zygmund inequality on a smooth simple arc
The classical Marcinkiewicz-Zygmund inequality asserts that, for \(T_n\) an arbitrary trigonometric polynomial of degree at most \(n\), one has \[ c_1|T_n |_p\leq \Biggl\{ {1\over {2n+1}} \sum_{k=0}^{2n} |T_n (x_k) |^p \Biggr\}^{1/p} \leq c_2|T_2 |_p, \] where \(x_k= 2k\pi/ (2n+1)\), \(k=0,1, \dots, 2n\), and the constants \(c_1\), \(c_2\) depend only on \(p\in (1, \infty)\). In this paper, similar inequalities are proved for \(P_{n-1}\) an arbitrary algebraic polynomial of degree at most \(n-1\) on a \(C^{2+ \delta}\) smooth simple arc \(\Gamma\). More precisely, there exist points \(z_{0,n}, z_{1,n}, \dots, z_{n,n}= z_{0,n}\) on \(\Gamma\) such that \[ c_3|P_{n-1} |_{L^p (\Gamma)}\leq \Biggl\{ \sum^n_{k=0} |P_{n-1} (x_k) |^p |z_{k+1,n}- z_{k,n}|\Biggr\}^{1/p}\leq c_4|P_{n-1} |_{L^p (\Gamma)}, \] where again the constants depend only on \(p\) and \(\Gamma\). From this one obtains theorems on the \(L^p\) convergence to \(f\) of some interpolation polynomials at the points \(z_{k,n}\), in the cases \(f\in C(\Gamma)\) and \(f\in L^p (\Gamma)\), as well as an estimate for the \(L^\infty\) approximation.
scientific article; zbMATH DE number 810858