Consider the analysis of a vector function ${\bf F}(\theta,\phi)$ in terms of the vector spherical harmonic (VSH) basis ${\bf Y}_{jm}^{l}(\theta,\phi)$ where $l=j-1,j,j+1$, and $j=0,\ldots, \infty$ and $m=-j, \ldots, j$. My question is: Do the vector spherical harmonics of equal $j$ and $m$ contribute equally to the angular spatial resolution in the analysis of ${\bf F}(\theta,\phi)$ ?

Here are a the $l=j-1,j,j+1$ vector spherical harmonics:

${\bf Y}_{jm}^{j+1}(\theta,\phi) = \sqrt{\frac{j+1}{2j+1}} \left(-{\bf e}_{r} Y_{jm}(\theta,\phi) + {\bf e}_{\theta} \frac{1}{j+1} \frac{\partial Y_{jm}(\theta,\phi)}{\partial \theta} + {\bf e}_{\phi} \frac{im}{j+1} \frac{Y_{jm}(\theta,\phi)}{\sin \theta} \right)$

${\bf Y}_{jm}^{j}(\theta,\phi) = -{\bf e}_{\theta} \frac{m}{\sqrt{j(j+1)}} \frac{Y_{jm}(\theta,\phi)}{\sin \theta} - {\bf e}_{\phi} \frac{i}{\sqrt{j(j+1)}} \frac{\partial Y_{jm}(\theta,\phi)}{\partial \theta}$

${\bf Y}_{jm}^{j-1}(\theta,\phi) = \sqrt{\frac{j}{2j+1}} \left({\bf e}_{r} Y_{jm}(\theta,\phi) + {\bf e}_{\theta} \frac{1}{j} \frac{\partial Y_{jm}(\theta,\phi)}{\partial \theta} + {\bf e}_{\phi} \frac{im}{j} \frac{Y_{jm}(\theta,\phi)}{\sin \theta} \right)$

Each of these VSHs of different $l$ depends upon scalar spherical harmonics of the same order in $j$ and $m$. But they also involve other functions of $\theta$ or derivatives with respect to $\theta$. Hence my question. With respect to $l=j+1$ and $l=j-1$ the angular spatial resolution provided by each is clearly equal since they both involve the same functions of $\theta$ and same derivative. But what about the $l=j$ VSH? It doesn't even have a component in the radial direction. So what can we say about the angular spatial resolution it contributes to the analysis. Can it be said to contribute equal resolution to that of the $l=j+1$ and $l=j-1$ VSHs?


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