Kähler information manifolds of signal processing filters in weighted Hardy spaces

We generalize K\"ahler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and composite functions of smooth transformations and transfer functions. We prove that the Riemannian geometry of a linear filter induced from weighted Hardy norms for the smooth transformations of its transfer function is the K\"ahler manifold. Additionally, the K\"ahler potential of the linear system geometry corresponds to the square of the weighted Hardy norms of its composite transfer functions. By using the properties of K\"ahler manifold, geometric objects on the manifolds from arbitrary weight vectors are computed in much simpler ways. Moreover, K\"ahler information manifolds of signal filters in weighted Hardy spaces generate various well-known information manifolds by the unified framework. We also cover several examples from time series models of which metric tensor, Levi-Civita connection, and K\"ahler potentials are represented with polylogarithm of poles and zeros from the transfer functions with the weight vectors are given as a family of exponential forms.