Some examples in degree of approximation by rational functions

Abstract

We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of E E , (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.