Recursive bi-orthogonalisation approach and orthogonal projectors

An approach is proposed which, given a family of linearly independent functions, constructs the appropriate bi-orthogonal 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 bi-orthogonal functions avoiding the need of inverse operations ii) it allows to quickly update a whole family of bi-orthogonal 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.