Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications

TL;DR

This paper generalizes angles and orthogonality concepts from Hilbert spaces to Banach spaces using an $L^p$ Pythagorean theorem, with applications in signal processing and preconditioning for large linear systems.

Contribution

It introduces a novel framework extending classical geometric notions to Banach spaces, enabling new applications in signal processing and numerical linear algebra.

Findings

Established an $L^p$ Pythagorean theorem in Banach spaces.

Developed new classes of preconditioners for large linear systems.

Provided numerical experiments demonstrating the framework's effectiveness.

Abstract

In the current work, we propose a generalization of angles and orthogonality from $L^2$ to generic Banach spaces, starting from a $L^p$ version of the Pythagorean theorem, $p\in [1,\infty)$. The starting point is conservation of energy measured in $L^1$ norm, as it occurs when considering the intrinsic mode functions decomposition in signal processing. This conservation of energy measure in $L^1$ norm is exactly the $L^1$ Pythagorean theorem. Besides the theoretical analysis, we apply the new notions in the context of preconditioning for structured large linear systems, by obtaining new classes of preconditioners. The present work contains numerical experiments and various remarks on the possible use of the given framework.