Weighted Marcinkiewicz inequalities and boundedness of the Lagrange operator (Q2711298)

In this interesting and important paper, the authors establish forward and converse Marcinkiewicz inequalities associated with doubling and \(A_p\) weights, for both trigonometric and algebraic polynomials.NEWLINENEWLINENEWLINELet \(1< p< \infty\) and \(u:[0,2\pi]\to [0,\infty)\) be such that \(u^p\) is a doubling weight, in the sense used in harmonic analysis. Let \(A,B> 0\) and \(\{\theta_j\}^{2m+1}_{j=0}\) satisfy NEWLINE\[NEWLINE0= \theta_0< \theta_1< \theta_2<\cdots< \theta_{2m+1}= 2\piNEWLINE\]NEWLINE as well as the spacing condition NEWLINE\[NEWLINE{A\over m}\leq \theta_{j+1}- \theta_j\leq {B\over m},\quad j=0,1,2,\dots, 2m.NEWLINE\]NEWLINE Then for all trigonometric polynomials \(T\) of degree \(\leq m\), NEWLINE\[NEWLINE\sum^{2m}_{j=0} \lambda_m(u^p, \theta_j)|T(\theta_j)|^p\leq C \int^{2\pi}_0|Tu|^p.NEWLINE\]NEWLINE Here \(\lambda_m(u^p,\theta_j)\) is the Christoffel function for \(u^p\) evaluated at \(\theta_j\), and the constant \(C\) depends only on \(A\), \(B\) and the doubling constant for \(u\). (We may also allow \(T\) to have degree \(\leq\ell m\), for fixed \(\ell\)). The essential feature is that \(C\) is independent of \(m\) and \(T\).NEWLINENEWLINENEWLINEConversely, let \(u^p\) be an \(A_p\) weight -- again in the sense used in harmonic analysis. Then NEWLINE\[NEWLINE\int^{2\pi}_0|Tu|^p\leq C \sum^{2m}_{j=0} \lambda_m\Biggl(u^p, {2\pi j\over 2m+1}\Biggr)\Biggl|T\Biggl({2\pi j\over 2m+1}\Biggr)\Biggr|^p,NEWLINE\]NEWLINE where \(C\) is independent of \(m\) and \(T\). One may also allow more general points than \(\{{2\pi j\over 2m+1}\}^{2m}_{j=0}\) at the expense of increasing the number of points.NEWLINENEWLINENEWLINEThe authors present generalizations of these, and show how they may be used to obtain estimates for the error in Lagrange interpolation. They also provide analogues for algebraic polynomials.NEWLINENEWLINENEWLINEThis paper is essential reading for those interested in weighted approximation, interpolation, and quadrature. It is one of a series of impressive papers in which these and other authors have shown that it is possible to treat a very broad class of doubling and \(A_p\) weights in the same way as generalized Jacobi weights were once treated.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00011].