Optimal Quadrature Formulas for the Sobolev Space $H^1$

TL;DR

This paper investigates optimal quadrature formulas in Sobolev space $H^1$, analyzing node placement for Fourier coefficient computation, and finds that equidistant nodes are optimal only under certain conditions related to the number of nodes and frequency.

Contribution

The study derives general error formulas for weighted integrals in $H^1$, and characterizes when equidistant nodes are optimal or suboptimal for Fourier coefficient approximation.

Findings

Equidistant nodes are optimal if $n \\ge 2.7 |k| +1$.

Equidistant nodes $x_j = j/|k|$ are worst for Fourier coefficients.

Optimal node placement depends on the relation between $n$ and $k$.

Abstract

We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space $H^1([0,1])$. We obtain general formulas for the worst case error depending on the nodes $x_j$. A particular case is the computation of Fourier coefficients, where the oscillatory weight is given by $\rho_k(x) = \exp(- 2 \pi i k x)$. Here we study the question whether equidistant nodes are optimal or not. We prove that this depends on $n$ and $k$: equidistant nodes are optimal if $n \ge 2.7 |k| +1 $ but might be suboptimal for small $n$. In particular, the equidistant nodes $x_j = j/ |k|$ for $j=0, 1, \dots , |k| = n+1$ are the worst possible nodes and do not give any useful information. To characterize the worst case function we use certain results from the theory of weak solutions of boundary value problems and related quadratic extremal problems.