Improving Gauss-Legendre

Technical Note — 18 Jul 2015

• signals on $\mathbb{S}^2$ always bandlimited to $L$, i.e., beyond $\ell\in\{0,1,\dotsc,L-1\}$ the harmonics are zero
• natural Gauss-Legendre uses $L$ samples along co-latitude ($\theta$) taken as the roots of the Legendre polynomial of degree $L$, $P_{L}(\theta)$ – denote these roots as $\theta^{\,r}_L$, $r=1,2,\dotsc,L$.
• when coupled with $2L-1$ samples along longitude ($\phi$) for each co-latitude sample, this sampling set is denoted $\mathfrak{S}_{\mathrm{GL}}$ and its cardinality is $\lvert\mathfrak{S}_{\mathrm{GL}}\rvert=L(2L-1)$

$$\begin{align*} \mathfrak{S}_{\mathrm{GL}} &= \Theta_L \times \Phi_{2L-1}, &\lvert\mathfrak{S}_{\mathrm{GL}}\rvert &= L(2L-1) \\ \mathfrak{S}_{\mathrm{EGL}} &= \{\Theta_{L-1} \times \Phi_{2L-1}\} \cup \{(0,0)\}, &\lvert\mathfrak{S}_{\mathrm{EGL}}\rvert &= (L-1)(2L-1)+1 \end{align*}$$

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Code Index:

• [06 Sep 2016] — Latex labels with MATLAB®
• [21 Jul 2015] — TikZ–tikzexternalize to png
• [18 Jul 2015] — Spherical Harmonics LaTeX Macros
• [01 Aug 2015] — BibDesk Publication html Export

Sphere Index:

• [06 Aug 2015] — Spherical Harmonic Plotting with MATLAB®
• [18 Jul 2015] — Spherical Harmonics LaTeX Macros
• [16 Jul 2015] — Spherical Harmonic MATLAB® Code 1

Misc Index:

• [06 Aug 2017] — HLab–1
• [31 Jul 2015] — Outputing png from pgfornament
• [18 Jul 2015] — Improving Gauss-Legendre