“Das letzte Ziel ist immer die Darstellung einer Funktion”: Grundlagen der analysis bei Weierstraß 1886, historische Wurzeln und Parallelen

Abstract

Weierstrass regarded his famous approximation theorem of 1885 primarily as a result on representations. In his lectures of 1886 he aimed at showing that even some classes of discontinuous functions can be represented by “arithmetical expressions,” i.e., series of polynomials. Also, and with some surprise, he had learned from the work of Fourier that fairly arbitrary functions could appear as boundary or initial distributions of analytic solutions to differential equations. In his eyes, all this justified the Eulerian conception of a function as represented by an expression. Weierstrass was apparently unaware of related methods and results by Cauchy and his contemporaries. In particular, Weierstrass' use of “delta functions” and convergence factors looks clumsier than earlier work done at Paris around 1820. The limited knowledge and mention of early 19th-century real analysis in the lectures of 1886 and its contrast to verbal praise of the significance of history, may explain a strange fact in historiography: the Weierstrassians show a lack of understanding, or even of knowledge, of earlier French real analysis.