On Converse Marcinkiewicz—Zygmund Inequalities in L p ,p>1

Abstract

We obtain converse Marcinkiewicz—Zygmund inequalities such as $$ \| P\nu \|_{L_{p}[-1,1]}\leq C\left( \sum_{j=1}^{n}\mu _{j}| P(t_{j})| ^{p}\right) ^{1/p} $$ for polynomials P of degree ≤ n-1 , under general conditions on the points {t j } n j=1 and on the function ν . The weights {μ j } n j=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [-1,1] .