# Computation of Induced Orthogonal Polynomial Distributions

## TL;DR

This paper introduces a stable, spectrally-accurate algorithm for computing distribution functions of induced orthogonal polynomial measures, enabling efficient sampling crucial for advanced polynomial approximation methods.

## Contribution

The authors present a novel, robust algorithm for computing and sampling from induced orthogonal polynomial distributions, with implementation available as open-source code.

## Key findings

- Stable computation for polynomial degrees up to 1000
- Efficient sampling method for induced distributions
- Application to multivariate polynomial approximation

## Abstract

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, which is stable for polynomial degrees up to at least degree 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.