The solution of the general interpolation problem has very many applications in numerical analysis and applied mathematics. However, some time ago, I realized that its possibilities have not been fully exploited and have even been underestimated, and I began to work on the subject. The concept underlying the problem is that of biorthogonality which gave its title to this book. It has many unusual connections and applications to Fourier expansion, projections, divided differences, extrapolation processes, numerical methods for integrating differential equations or for solving integral equations, rational approximations to formal power series and series of functions, least squares, statistics, and biorthogonal polynomials, to name some. Most of the results given in this book are new and have not even been published in the form of journal articles. They appear here for the first time. This is the case in particular for the various recurrence relations given and for the generalizations of the method of moments, the method of Lanczos, and the biconjugate gradient method. New approximations of Pade-type for series are also described. The possibilities opened by the concept of biorthogonality have still to be discovered and new applications as well. Thus, this book will be of interest to researchers in numerical analysis and approximation theory. However, this does not mean that the material given here is difficult. Almost no prerequisite are needed and the book can also be used as a text for students.
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