Birkhoff, or lacunary, interpolation appears whenever observation gives scattered, irregular information about a function and its derivatives. First discovered by G. D. Birkhoff in 1906, it received little attention until I. J. Schoenberg revived interest in the subject in 1966. Lacunary interpolation differs radically from the more familiar Lagrange and Hermite interpolation in both its problems and its methods. It could even be described as “non-Hermitian” interpolation. The name Birkhoff interpolation is justified also from a historical point of view.

At present, the main definitions and theorems for polynomial Birkhoff interpolation seem to have been found, while the theory for other systems of functions, most notably splines, is in healthy development. Since this material can be found only in research periodicals and proceedings of conferences, it is time for a comprehensive exposition of the material. We have gone to great lengths to unify, simplify, and improve the information already published or in press, and to set the stage for further developments.

The book should be of interest to approximation theorists, numerical analysts, and analysts in general, as well as to computer specialists and engineers who need to analyze functions when their values and those of their derivatives are given in an erratic way. The book could be used as a text for a graduate course, requiring little more than an undergraduate mathematics background.

Many novel ideas and tools have been developed in this theory; interpolation matrices, coalescence of rows in matrices, independent knots, probabilistic methods, diagrams of splines, and the Rolle theorem for splines.