Weierstrass and Uniform Approximation

We present the theorem of uniform approximation of continuous functions by polynomials in the Weierstrass framework for the construction of analysis based on the representation of functions by sums of power series and analytic functions, and his effort to bring new standards of rigor to analysis that accompanied his differences with Riemann. We also describe the most original proofs of the approximation theorem that appeared after the Weierstrass death. 1. Karl Weierstrass (1815 Ostenfelde-1897 Berlin) Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms. Encyclopaedia Britannica While his studies at the Catholic Gymnasium in Paderborn, Weierstrass regularly read Crelle’s Journal and decided to devote himself to Mathematics. But in 1834, when he was 19 years old, his domineering father, who was a tax inspector, sent him to the University of Bonn to study finance and laws. But not having any interest for these studies, in Bonn he spend time in amusement, and also reading Laplace’s “Mécanique céleste” and Jacobi’s work “Fundamenta nova theoriae functionum ellipticarum”, on a topic that was very popular at the time. This work, based of the recently published “Trait des fonctions elliptiques” by Legendre, proved difficult for him. To learn the techniques and methods in elliptic functions, he read a transcription of a lecture on modular functions by Christoph Gudermann, a former student of Gauss that later became Weierstrass’ advisor. As a result, four years later, he left the University with no degree, and the conflict between his academic duties and mathematical inclinations led to mental and physical strain. 1Crelle is the comon name of the “Journal für die reine und angewandte Mathematik”. Founded by August Crelle in 1826 and edited by him until his death in 1855, it was the first major mathematical journal that was not a proceedings of an academy. Weierstrass and Kronecker were the editors from 1881 to 1888.