Recursive bi-orthogonalisation approach and orthogonal projectors

TL;DR

This paper introduces an iterative method for constructing biorthogonal functions for a given set, enabling efficient representation of functions as superpositions of non-orthogonal waveforms without inverse operations.

Contribution

It presents a recursive bi-orthogonalisation approach that efficiently updates biorthogonal functions when new elements are added, avoiding inverse calculations.

Findings

Provides a fast, iterative procedure for biorthogonal set construction

Enables efficient updates when adding new functions

Facilitates function approximation with non-orthogonal waveforms

Abstract

An approach is proposed which, given a family of linearly independent functions, constructs the appropriate biorthogonal set so as to represent the orthogonal projector operator onto the corresponding subspace. The procedure evolves iteratively and it is endowed with the following properties: i) it yields the desired biorthogonal functions avoiding the need of inverse operations ii) it allows to quickly update a whole family of biorthogonal functions each time that a new member is introduced in the given set. The approach is of particular relevance to the approximation problem arising when a function is to be represented as a finite linear superposition of non orthogonal waveforms.