Orthogonality and Biorthogonality: theory, numerical algorithms and applications

The notion of biorthogonality is a very general one having applications in many branches of numerical analysis. For example, it leads, on one hand, to formal biorthogonal polynomials and, on the other hand, to the methods of moments of Vorobyev and the method of Galerkin.

Formal biorthogonal polynomials open the way to the concepts of formal vector and scalar orthogonality which form a natural basis for the study of vector and scalar Padé approximants, continued fractions, rational interpolation, Gaussian quadratures and the Kronrod procedure for estimating its error, projection methods for solving systems of linear equations, and extrapolation methods for scalar and vector sequences. The concept could be extended to matrix orthogonality which is already used for block systems of linear equations. Methods for the solution of systems of nonlinear equations also benefit from this approach.

The method of moments leads to Galerkin method and to projection methods for solving systems of linear equations (Krylov subspace methods and their generalizations), and the solution of the matrix eigenvalue problem (Lanczos-type methods). Some convergence acceleration methods recently used in the Google’s PageRank problem for web search have also been put into this framework.

An important role in the theory of biorthogonality is played by ratios of determinants, recursive computational rules, and the notion of Schur complement which have been extensively reviewed in a book published in 2005. Their respective roles have to be emphasized.

Another field of interest concerns biorthogonal wavelets which are used in signal and image processing. The general theory of biorthogonality could lead to a general framework for such wavelets, and to new computational procedures using, for example, the recursive projection algorithms introduced in 1983.

Biorthogonality, and, in particular, formal orthogonal polynomials, have been studied for many years now, and they play a crucial role in various important numerical methods and applications. It is now time to gather and extend these results in various directions, and this is one of the main goal of this project.

Theoretical and computational issues concerning multivariate OP (Orthogonal Polynomials) are an active research area, especially from the '90s, also due to their connections with other fields of multivariate approximation theory like interpolation and cubature and their applications. Compared to the univariate case, the field can be considered still far from its maturity, in particular concerning computational methods.

A notion, closely related to biorthogonality is the so called ``quasi-orthogonality''. Quasi-orthogonal polynomials are a generalization of orthogonal polynomials and some properties were studied in the univariate case, determining the locations of their zeros and the measure of orthogonality. They have applications in Gauss-Radau and Gauss-Lobatto quadrature formulae. Nevertheless, quasi-orthogonal polynomials on the unit circle and the multivariate case have not been much investigated.

Michela Redivo Zaglia