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.on a closed point set of the extended plane, for as we shall now prove, approximation on a set C which is not closed can be reduced to approximation on the set C composed of C and its limit points. Instead of considering approximation, it isanbsp;...
|Title||:||Interpolation and Approximation by Rational Functions in the Complex Domain|
|Author||:||J. L. Walsh|
|Publisher||:||American Mathematical Soc. - 1935-12-31|