removing manual spacing

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