A fast procedure for the construction of quadrature formulas for bandlimited functions

TL;DR

This paper presents a fast, efficient method for constructing quadrature formulas for bandlimited functions, leveraging prolate spheroidal wave functions with an asymptotic CPU time of approximately O(n log n).

Contribution

It introduces a novel, practical algorithm for quadrature rule construction that significantly reduces computational complexity compared to traditional methods.

Findings

CPU time for construction is approximately O(n log n)

Algorithm performs well in numerical examples

Practical for large n due to small constant factors

Abstract

We introduce an efficient scheme for the construction of quadrature rules for bandlimited functions. While the scheme is predominantly based on well-known facts about prolate spheroidal wave functions of order zero, it has the asymptotic CPU time estimate $O(n log n)$ to construct an n-point quadrature rule. Moreover, the size of the ``$n log n$'' term in the CPU time estimate is small, so for all practical purposes the CPU time cost is proportional to $n$. The performance of the algorithm is illustrated by several numerical examples.