I am interested in interpolation using spherical harmonics.
I feel like I searched all Google pages containing key words of this subject. Therefore I want to ask if anyone is familiar with this, and could recommend material to me (like papers, books or webpages) where I can read about this.
Thanks!
The last chapter or two of Dym and McKean's Fourier Series and Integrals addresses this to some degree, but it's heavy going.
In general, if you want to approximate a function $f: S^2 \subset \mathbb R^3 \to \mathbb R$ by a sum $$ f(x, y, z) \approx \sum c_i h_i(x, y, z) $$ you can find the coefficients $c_i$ by computing $$ c_i = \int_{S^2} f(x, y, z) h_i(x, y, z) ~dA $$ just as you compute Fourier coefficients by integrating over the circle. This works because the harmonics $h_i$ are an orthogonal set of functions (with respect to the inner product "integrate $fg$ over $S^2$"); you're really just doing orthogonal projections onto the axes of a subspace of the space of all integrable functions on the sphere.