Sampling, Marcinkiewicz-Zygmund Inequalities, Approximation, and Quadrature Rules

TL;DR

This paper develops approximation theorems and quadrature rules based on Marcinkiewicz-Zygmund inequalities in L^2, using elementary methods involving Sobolev spaces and least squares solutions.

Contribution

It introduces new approximation and quadrature results derived solely from the basic properties of Marcinkiewicz-Zygmund inequalities, Sobolev spaces, and least squares.

Findings

Derived approximation theorems in L^2

Constructed quadrature rules from inequalities

Method is elementary and broadly applicable

Abstract

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L^2$, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.