# An O(1) Algorithm for the Numerical Evaluation of the Prolate Spheroidal Wave Functions of Order 0

## TL;DR

This paper introduces an O(1) time algorithm for numerically evaluating prolate spheroidal wave functions of order 0, significantly improving efficiency for large parameters and enabling faster computations.

## Contribution

The authors develop a novel O(1) algorithm for evaluating prolate spheroidal wave functions, independent of order and bandlimit parameters, which is a substantial advancement over existing methods.

## Key findings

- Algorithm runs in constant time regardless of parameters
- Numerical experiments confirm accuracy and efficiency
- Implementation is publicly available for use

## Abstract

The standard algorithm for the numerical evaluation of the prolate spheroidal wave function $\mathsf{Ps}\hskip.05em{}_{n}(x;\gamma^2)$ of order $0$, bandlimit $\gamma > 0$ and characteristic exponent $n$ has running time which grows with both $n$ and $\gamma$. Here, we describe an alternate approach which runs in time independent of these quantities. We present the results of numerical experiments demonstrating the properties of our scheme, and we have made our implementation of it publicly available.