I am working on a data analysis project, as a part of which I want to express a probability distribution as a spherical harmonic expansion on a 2-sphere. Imposing the condition of realness of the function gives some nice, convenient conditions on the spherical harmonic coefficients as below.
$$ a_{l, -m} = (-1)^m \, a^*_{lm} \; \; \; \; \text{where} \; \; \; \; \; f(\theta, \phi) = \sum_{l, m} a_{lm} Y_{lm} (\theta, \phi) $$
The probability distribution is also non-negative, in addition to being real. So I was curious if there are any additional conditions which we can get from that restriction. I wasn't able to work out something myself and my google-fu didn't get me anything relevant.
Any help is really appreciated! Apart from just my curiosity - further conditions on $a_{lm} $ would also be helpful in reducing the dimensionality of the space of the data analysis work. Thanks!