Wirtinger's Calculus in general Hilbert Spaces

TL;DR

This paper provides a rigorous extension of Wirtinger's calculus to general Hilbert spaces, including Reproducing Kernel Hilbert Spaces, aiming to deepen theoretical understanding and facilitate advanced applications.

Contribution

It introduces a more rigorous and insightful presentation of Wirtinger's calculus and extends its framework to encompass general Hilbert spaces and kernel methods.

Findings

Extended Wirtinger's calculus to Reproducing Kernel Hilbert Spaces

Provided rigorous theoretical foundations for the calculus in Hilbert spaces

Facilitated potential applications in kernel-based learning algorithms

Abstract

The present report, has been inspired by the need of the author and its colleagues to understand the underlying theory of Wirtinger's Calculus and to further extend it to include the kernel case. The aim of the present manuscript is twofold: a) it endeavors to provide a more rigorous presentation of the related material, focusing on aspects that the author finds more insightful and b) it extends the notions of Wirtinger's calculus on general Hilbert spaces (such as Reproducing Hilbert Kernel Spaces).