Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc (2410.22045v1)

Abstract: We investigate an extended version of Hilbert space of analytic functions called Hilbert space of complex-valued harmonic functions. It is found that functions in Hilbert space of complex-valued harmonic functions exhibit many properties analogous to its analytic counter part such as complex-valued harmonic function analogous of norm, equivalent norms, reproducing kernels, growth estimates and Littlewood-Paley Identity Theorem. In conclusion we prove that many results in Hilbert space of analytic functions also hold in larger Hilbert space of complex-valued harmonic functions.