On exact discretization of the $L_2$-norm with a negative weight

TL;DR

This paper investigates the minimal number of points needed for exact discretization of the $L_2$-norm in a function subspace, demonstrating the necessity of negative weights in some cases.

Contribution

It constructs a subspace where any exact discretization requires at least one negative weight, highlighting limitations of positive-weight discretizations.

Findings

Existence of subspaces requiring negative weights for exact discretization

Lower bounds on the number of nodes for discretization

Implications for numerical integration and approximation theory

Abstract

For a subspace $X$ of functions from $L_2$ we consider the minimal number $m$ of nodes necessary for the exact discretization of the $L_2$-norm of the functions in $X$. We construct a subspace such that for any exact discretization with $m$ nodes there is at least one negative weight.