Orthonormality of spherical basis functions for interior problems of the Helmholtz equation

We construct an orthonormal basis for interior problems of the Helmholtz equation, based on the properties of a reproducing kernel Hilbert space defined by the spectral characteristics of interior sound fields. The constructed basis coincides with what is commonly known as spherical basis functions. Furthermore, leveraging the structure of this space, we derive the addition theorem in a compact form. This facilitates the conversion between reproducing kernel representations and spherical harmonic expansions and provides insights into estimating spherical harmonic coefficients from sampled measurements.