Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants

TL;DR

This paper introduces a Wronski determinant-based method for constructing linearly independent and orthogonal functions in Hilbert spaces, generalizing Gram-Schmidt, with applications to differential equations and basis functions.

Contribution

It presents a novel, general approach using Wronski determinants to build orthogonal functions in Hilbert spaces, extending classical methods.

Findings

Method successfully constructs orthogonal functions from a single initial function.

Applications include solving differential equations and creating basis functions.

Proposes a conjecture linking basis functions and Wronski determinants.

Abstract

Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that satisfies mild conditions, and emerges as a generalization of the Gram-Schmidt process. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to the introduction of the Wronski basis.