Suppose there is a function $f(\theta,\phi)$ defined on the surface of a sphere, and $\theta$ and $\phi$ are the polar and azimuthal angles respectively. Similar with the fact that a function defined on a line or a circle can be expanded with respect to the Fourier bases, this function $f$ can be expanded in terms of the spherical harmonics $Y_l^m(\theta, \phi)$: $$f(\theta,\phi)=\sum_{l=0}^{\infty} \sum_{m=-l}^l F_l^m Y_l^m(\theta,\phi),$$ where $l$ and $m$ are the degree and order of the harmonics. In order for the expansion to achieve a perfect fitting to the original function, $l$ needs to go from 0 to $\infty$.
In our experiment design, however, we cannot detect directly the value of the original function but can only detect the expansion coefficients $F_l^m$, and unfortunately, the orders of $F_l^m$ we are able to detect is limited, ie. $l \leq l_0$, so in fact we are using a truncated summation $$\sum_{l=0}^{l_0} \sum_{m=-l}^l F_l^m Y_l^m(\theta,\phi)$$ to simulate the original function. When $l_0$ is small, even though the truncated function captures the overall variation trends of the original function well, the amplitudes are significantly different from the originals.
So my question is: is there a way to mathematically scale the truncated function value to a direct comparable level with the original function? If so, how to do it please? Any input would be greatly appreciated, thanks a lot!