Kähler information manifolds of signal processing filters in weighted Hardy spaces
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We generalize Kähler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and generic composite functions of transfer functions. We prove that the Riemannian geometry induced from weighted Hardy norms for composite functions of its transfer function is the Kähler manifold. Additionally, the Kähler potential of the linear system geometry corresponds to the square of the weighted Hardy norms for composite functions of its transfer function. By using the properties of Kähler manifolds, it is possible to compute various geometric objects on the manifolds from arbitrary weight vectors in much simpler ways. Additionally, Kähler information manifolds of signal filters in weighted Hardy spaces can generate various information manifolds such as Kählerian information geometries from the unweighted complex cepstrum or the unweighted power cepstrum, the geometry of the weighted stationarity filters, and mutual information geometry under the unified framework. We also cover several examples from time series models of which metric tensor, Levi-Civita connection, and Kähler potentials are represented with polylogarithm of poles and zeros from the transfer functions when the weight vectors are in terms of polynomials.
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