Notes on inequalities with doubling weights
Weighted trigonometric polynomial polynomial approximation on a finite interval is a central theme in approximation theory. For many years, the weights for which analysis is possible were restricted to the generalized Jacobi weights and their cousins. A recent breakthrough, due to Mastroianni and Totik, is that analysis is possible for doubling weights. Recall that a \(2\pi\)-periodic function \(W:\mathbb{R}\to [0,\infty)\) is called a doubling weight, if there exists \(L>0\) such that for every interval \(I\), \(W(2I)\leq LW(I)\), where \(2I\) is the interval with the same midpoint as \(I\), but twice the length, and \(W(I)= \int_IW\). Mastroianni and Totik established weighted inequalities of Bernstein, Marcinkiewicz, Nikolskii, Remez and Schur, as well as estimates for Christoffel functions, in an \(L_p\) setting, with \(1\leq p<\infty\). In this interesting paper, the author extends these to the case \(0<p<1\), using an estimate for quadrature sums, due to Máté, Nevai, and the reviewer. The extension is by no means straightforward. For example, the Bernstein inequality reads as follows: let \(W\) be a doubling weight and \(0<p <\infty\). There exists \(C>0\) depending only on \(p\) and the doubling constant \(L\) such that for \(n\geq 1\) and all trigonometric polynomials \(T_n\) of degree \(\leq n\), \[ \int^\pi_{-\pi} |T_n'|^p W_n\leq Cn^p\int^\pi_{-\pi} |T_n|^pW. \] Here \[ W_n(x)= n\int^{x+ {1\over n}}_{x- {1\over n}}W. \]
scientific article; zbMATH DE number 1356331