Defining family of linear maps $[F]_{l,m}$ as $$ [F]_{l,m} = \int_{S_1} F(\hat n) \exp(ik \hat r\cdot \hat n) Y_l^m dS(\hat n) $$ (the integral is over the unit sphere)

This map is interesting in the analysis of the wave equation in spherical coordinate systems. Now using the identity (https://en.wikipedia.org/wiki/Plane_wave_expansion) $$ \exp(i k \hat r \cdot \hat n) = 4 \pi \sum_{l=0}^\infty \sum_{m=-l}^{l} i^l j_l(k r) (Y_l^m(\hat n))^* Y_l^m(\hat r) $$ ($j_i$ spherical bessel function) and that the spherical harmonics $Y_{l,m}$ is orthogonal, then $$ [1]_{l,m} = C j_l(k r) Y_l^m $$ (C constant) Now any derivative of this will result in a derivative of spherical bessel functions which each is a linear combination of spherical bessel functions. Also we have formally $$ \frac{\partial}{\partial x_i}[F]_{l,m} = [i k n_i F]_{l,m} $$

So if we assume $F$ in a suitable function class, say all polynomials under the complex numbers, then we can define $[F]'_{i,j}$ as taking $[F]_{l,m}$ and replace all $j_i$ with $y_i$ e.g. the second solution to the spherical bessel equation (https://en.wikipedia.org/wiki/Bessel_function). $[],[]'$ share a lot of similar properties but my main task is to see if there exists a way to characterize $[]'$ as an integral over a kernel and what that kernel is e.g i want to define $[]'_{l,m}$ as $$ [F]'_{l,m}(\hat r) = \int F(\hat z) K_{l,m}(\hat z,\hat r)d\mu_{l,m}(\hat z) ?. $$

One approach of a characterization is to use orthogonality and let $H_{l,m}(\hat r, \hat n)$ be defined by

$$ H_{l,m}(\hat r, \hat n) = 4 \pi \sum_{l=0}^\infty \sum_{m=-l}^{l} i^l y_l(k r) (Y_l^m(\hat n))^* Y_l^m(\hat r) $$

Then as with $[]_{l,m}$, we define $$ [F]_{l,m}' = \int_{S_1}F(\hat n) H_{l,m}(\hat r,\hat n) Y_l^m dS(\hat n)? $$

But this is cheating it's not a proven that this will represent the map. My question is if there is a simple expression for $K_{l,m},\mu_{l,m}$, just as there exists a simple analytical expression for the kernel of $[]_{l,m}$.



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