From bc601d7493c6eb62b4cce2c012ca4dd44c4f4947 Mon Sep 17 00:00:00 2001 From: Boris Bonev Date: Tue, 24 Oct 2023 13:15:36 +0200 Subject: [PATCH] removing manual spacing --- README.md | 12 ++++-------- 1 file changed, 4 insertions(+), 8 deletions(-) diff --git a/README.md b/README.md index 3b31788..671f1b5 100644 --- a/README.md +++ b/README.md @@ -116,22 +116,18 @@ $$ f_l^m = \int_{S^2} \overline{Y_l^m}(\theta, \lambda) \, f(\theta, \lambda) \; \mathrm{d} \mu(\theta, \lambda) $$ -realizes the projection of a signal $f(\theta, \lambda)$ on $S^2$ onto the spherical harmonics basis. - -generalizes the Fourier transform on the sphere. - -The truncated series expansion of a function $f$ defined on the surface of a sphere can be written as +realizes the projection of a signal $f(\theta, \lambda)$ on $S^2$ onto the spherical harmonics basis. The SHT generalizes the Fourier transform on the sphere. Conversely, a truncated series expansion of a function $f$ can be written in terms of spherical harmonics as $$ -\hat{f} (\theta, \lambda) = \sum_{m=-M}^{M} \exp(im\lambda) \sum_{l=|m|}^{M} \hat{f}_l^m \overline{P_l^m} (\cos \theta), +f (\theta, \lambda) = \sum_{m=-M}^{M} \exp(im\lambda) \sum_{l=|m|}^{M} \hat{f}_l^m \overline{P_l^m} (\cos \theta), $$ -where $\theta$ is the colatitude, $\lambda$ the longitude, $\overline{P_l^m}$ the normalized, associated Legendre polynomials and $F_n^m$, the expansion coefficient associated to the mode $(m,n)$. +where $\hat{f}_l^m$, are the expansion coefficients associated to the mode $m,n$. The implementation of the SHT follows the algorithm as presented in [2]. A direct spherical harmonic transform can be accomplished by a Fourier transform $$ -\hat{f}^m(\theta) = \frac{1}{2 \pi} \int_{0}^{2\pi} f(\theta, \lambda) \, \exp(-im\lambda) \; \mathrm{d}\lambda +\hat{f}^m(\theta) = \frac{1}{2 \pi} \int_{0}^{2\pi} f(\theta, \lambda) \exp(-im\lambda) \mathrm{d}\lambda $$ in longitude and a Legendre transform