REPRESENTATION THEOREM FOR QUATERNIONIC HARDY SPACES VIA SLICE HYPERHOLOMORPHIC FUNCTIONS

Abstract

We establish a novel representation theorem for quaternionic Hardy spaces defined via slice hyperholomorphic functions on the unit ball of quaternions. This work extends the classical Szegő kernel representation from complex analysis to the quaternionic setting, utilizing the framework of slice hyperholomorphicity. The main results demonstrate that the quaternionic Hardy space forms a right quaternionic Hilbert space with a reproducing kernel and integral representation analogous to the complex case. These findings provide a foundation for operator theory in quaternionic Hilbert spaces and suggest potential applications in signal processing and mathematical physics.