# Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series

## TL;DR

This paper introduces a fast, stable method for transforming between spherical harmonic expansions and bivariate Fourier series, utilizing butterfly factorizations and low-rank matrix decompositions for efficiency.

## Contribution

It presents a novel two-step transformation process with provable optimal asymptotic complexity, combining butterfly factorization and hierarchical low-rank techniques.

## Key findings

- Total pre-computation complexity is rac{rac{O(n^3 \, log n)}{}}
- Execution time is asymptotically rac{rac{O(n^2 \, log^2 n)}{}}
- Method is numerically stable and rigorously analyzed

## Abstract

A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all orders are converted to those of order zero and one; then, these intermediate expressions are re-expanded in trigonometric form. The first step proceeds with a butterfly factorization of the well-conditioned matrices of connection coefficients. The second step proceeds with fast orthogonal polynomial transforms via hierarchically off-diagonal low-rank matrix decompositions. Total pre-computation requires at best $\mathcal{O}(n^3\log n)$ flops; and, asymptotically optimal execution time of $\mathcal{O}(n^2\log^2 n)$ is rigorously proved via connection to Fourier integral operators.