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<p>We consider optimal interpolation of functions analytic in simply connected domains in the complex plane. By choosing a specific structure for the approximant, we show that the resulting first-order optimality conditions can be interpreted as optimal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{\mathcal {H}}_{\varvec{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-script">H</mi> </mrow> <mrow> <mn mathvariant="bold">2</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> interpolation conditions for discrete-time dynamical systems. Connections to model reduction of discrete-time time-invariant delay systems are also established with particular emphasis on discretized linear systems obtained through the implicit Euler method, the midpoint method, and backward differentiation methods. A data-driven algorithm is developed to compute a (locally) optimal approximant. Our method is tested on three numerical experiments.</p>

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Data-driven optimal approximation on Hardy spaces in simply connected domains

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月9日

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  • Alessandro Borghi,
  • Tobias Breiten

摘要

We consider optimal interpolation of functions analytic in simply connected domains in the complex plane. By choosing a specific structure for the approximant, we show that the resulting first-order optimality conditions can be interpreted as optimal \(\varvec{\mathcal {H}}_{\varvec{2}}\) H 2 interpolation conditions for discrete-time dynamical systems. Connections to model reduction of discrete-time time-invariant delay systems are also established with particular emphasis on discretized linear systems obtained through the implicit Euler method, the midpoint method, and backward differentiation methods. A data-driven algorithm is developed to compute a (locally) optimal approximant. Our method is tested on three numerical experiments.