We generalize K\"ahler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and smooth transformations of 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 a 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. Based on properties of K\"ahler manifolds, geometric objects on the manifolds of the linear systems in weighted Hardy spaces are computed in much simpler ways. Moreover, K\"ahler information manifolds of signal filters in weighted Hardy spaces incorporate various well-known information manifolds under 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 polylogarithms of poles and zeros from the transfer functions with weight vectors in exponential forms.

翻译：我们通过引入加权Hardy空间和传递函数的光滑变换，推广了复值信号处理滤波器的Kähler信息流形。我们证明，由传递函数光滑变换的加权Hardy范数所诱导的线性滤波器的黎曼几何结构构成一个Kähler流形。此外，该线性系统几何的Kähler势等于其复合传递函数加权Hardy范数的平方。基于Kähler流形的性质，加权Hardy空间中线性系统流形上的几何对象可以以更简单的方式计算。更进一步，加权Hardy空间中信号滤波器的Kähler信息流形将多种经典信息流形纳入统一框架。我们还涵盖了几个时间序列模型的实例，其中度量张量、Levi-Civita联络和Kähler势均由传递函数极点和零点（结合指数形式的权向量）的多对数函数表示。