Taylor Series are Limits of Legendre Expansions

Abstract

Next to a power series, the classical Legendre series offers the simplest method of representing a function using polynomial expansion means. In 1862, Neumann established results for complex Legendre expansions that are analogous to Taylor's Theorem and the Cauchy-Hadamard Formula for power series, the primary difference being that results are stated in terms of ellipses, as opposed to discs, of convergence. After a simple change of variable, the foci of these ellipses may vary, each leading to a modified Legendre expansion of the original function. Our main result is that as the foci of these ellipses tend to one another, the limit of the corresponding Legendre expansions is the Taylor series representation.