Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc

The study investigated 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 counterpart such as complex-valued harmonic function analogous of norm, equivalent norms, reproducing kernels, growth estimates and Littlewood-Paley Identity Theorem. In particular, the researchers established that several fundamental results known for Hilbert space of analytic functions naturally extend to this broader harmonic framework. Beyond theoretical interest, these findings provide new tools for studying operator theory, potential theory, and approximation processes within the harmonic setting, thereby opening avenues for further research and applications in related areas of Mathematics and applied sciences.