Beamforming in the reproducing kernel domain based on spatial differentiation

This paper proposes a novel beamforming framework in the reproducing kernel domain, derived from a unified interpretation of directional response as spatial differentiation of the sound field. By representing directional response using polynomial differential operators, the proposed method enables the formulation of arbitrary beam patterns including non-axisymmetric. This is achieved by reinterpreting beamforming as a functional acting on the sound field, parameterized by directivity function and a virtual observation position. The derivation of the reproducing kernel associated with the interior fields is mathematically supported by Hobson's theorem, which allows concise analytical expressions. Furthermore, the proposed framework generalizes conventional spherical harmonic (SH) domain beamformers by reinterpreting them as spatial differential operators, thereby clarifying their theoretical structure and extensibility. Four numerical simulations conducted in two- and three- dimensional space confirm the validity of the method. The results demonstrate improved sound field reconstruction accuracy when microphone directivity is explicitly modeled and show that the proposed beamformer achieves performance comparable to SH domain beamforming across a wide range of conditions while providing slightly improved performance at higher frequencies.