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

Abstract 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 $$\varvec{\mathcal {H}}_{\varvec{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> 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.