On Complex Analytic tools, and the Holomorphic Rotation methods

TL;DR

This paper discusses advanced complex analytic tools and holomorphic rotation methods, extending them from Hardy spaces in the upper half-plane to harmonic functions in higher dimensions, enabling new applications in multidimensional signal processing.

Contribution

It introduces transfer techniques for nonlinear analytic approximation tools from Hardy spaces to higher-dimensional harmonic functions, with new representation theorems and isometries.

Findings

Representation theorems for harmonic functions in higher dimensions

Extension of complex analytic tools to multidimensional settings

Potential applications in processing oscillatory multidimensional signals

Abstract

We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It is known [6] that all harmonic functions in higher dimensions are combinations of holomorphic functions on 2 dimensional planes, extended as, constant in normal directions. We derive representation theorems, with corresponding isometries, opening the door for applications in higher dimensions, to the processing of highly oscillatory multidimensional signals.