The present work is restricted to the representation of functions in the complex domain, particularly analytic functions, by sequences of polynomials or of more general rational functions whose poles are preassigned, the sequences being defined either by interpolation or by extremal properties (i.e. best approximation). Taylor's series plays a central role in this entire study, for it has properties of both interpolation and best approximation, and serves as a guide throughout the whole treatise. Indeed, almost every result given on the representation of functions is concerned with a generalization either of Taylor's series or of some property of Taylor's series--the title "Generalizations of Taylor's Series" would be appropriate.

Table of Contents

• Possibility of approximation, analytic functions
• Possibility of approximation, continued
• Interpolation and lemniscates
• Degree of convergence of polynomials. Overconvergence
• Best Approximation by polynomials
• Orthogonality and least squares
• Interpolation by polynomials
• Interpolation by rational functions
• Approximation by rational functions
• Interpolation and functions analytic in the unit circle
• Approximation with auxiliary conditions and to non-analytic functions
• Existence and uniqueness of rational functions of best approximation
• Appendix
• Bibliography
• Index