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

2017/05/15 by Richard Mikaël Slevinsky, Slevinsky, Richard Mikael · 2 citations
Computer Science · Earth and Planetary Sciences · Mathematics · #33C55 #65F99 #65Y99 #FOS: Mathematics #Geophysics and Gravity Measurements #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.1705.05448

openalex publication_date 2017/05/15 · openalex created_date 2022/10/06 · openalex updated_date 2026/08/01

Abstract

A rapid transformation is derived between spherical harmonic expansions and\ntheir analogues in a bivariate Fourier series. The change of basis is described\nin two steps: firstly, expansions in normalized associated Legendre functions\nof all orders are converted to those of order zero and one; then, these\nintermediate expressions are re-expanded in trigonometric form. The first step\nproceeds with a butterfly factorization of the well-conditioned matrices of\nconnection coefficients. The second step proceeds with fast orthogonal\npolynomial transforms via hierarchically off-diagonal low-rank matrix\ndecompositions. Total pre-computation requires at best \O(n³\log n)\nflops; and, asymptotically optimal execution time of \O(n²\log² n)\nis rigorously proved via connection to Fourier integral operators.\n

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• Nonlocal vector calculus on the sphere

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