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Marcinkiewicz-Zygmund Inequalities: Methods and Results

  • D. S. Lubinsky
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 430)

Abstract

The Gauss quadrature formula for a weight W 2 on the real line has the form
$$\sum_{j=1}^{n}\lambda _{jn}P(x_{jn})=\int PW^{2}$$
for polynomials P of degree ≤ 2n − 1. In studying convergence of Lagrange interpolation in L p norms, p ≠ 2, one needs forward and converse quadrature sum estimates such as
$$\sum_{j=1}^{n}\lambda _{jn}W^{-2}(x_{jn})|PW|^{p}(x_{jn})\leq \geq C\int |PW|^{p}$$
with C independent of n and P. These are often called Marcinkiewicz-Zygmund inequalities after their founders. We survey methods to prove these and the results that can be achieved using them. Our focus is on weights on the whole real line, but we also refer to results for (−1,1) and the plane. In particular, we present four methods to prove forward estimates and two to prove converse ones.

Key words and phrases

Quadrature sums Gauss quadrature Marcinkiewicz-Zygmund inequalities 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • D. S. Lubinsky
    • 1
  1. 1.Mathematics DepartmentWitwatersrand UniversityWitsSouth Africa

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