ABSTRACT
The notion of biorthogonality is obviously a generalization of the notion of orthogonality in an Hilbert space which itself comes from the notion of orthogonality for functions and polynomials. Chapter VII of Banach’s book of 1932 is devoted to the general notion of biorthogonality. Although biorthogonality received some attention since that time, it was only quite recently that the study of biorthogonal polynomials in connection with some problems in rational approximation and numerical methods for ordinary differential equations, appeared on the scene and played a central rôle (see, for example, [110]). Orthogonality of dimension d for polynomials [103] and, equivalently, 1/d-orthogonality [131] were recently the subjects of investigations and applications. All these new notions of orthogonality for polynomials are particular cases of the general notion of biorthogonality which also provides, as we shall see below, a natural and general framework for the definition and the study of generalizations of many concepts and methods such as the methods of moments and that of Galerkin, Lanczos’ bi-orthogonalization process, the bi-conjugate gradient method, projections, Padé approximants of various types, extrapolation methods for scalar and vector sequences, and so on.
