Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German mathematician who was born in Berlin and died in Freiburg. He was the brother of Emil du Bois-Reymond.
His thesis was concerned with the mechanical equilibrium of fluids. He worked on the theory of functions and in mathematical physics. His interests included Sturm–Liouville theory, integral equations, variational calculus, and Fourier series. In this latter field, he was able in 1873 to construct a continuous function whose Fourier series is not convergent. His lemma defines a sufficient condition to guarantee that a function vanishes almost everywhere.
In a paper of 1875, du Bois-Reymond employed for the first time the method of diagonalization, later associated with the name of Cantor.[1] Du Bois-Reymond also established that a trigonometric series that converges to a continuous function at every point is the Fourier series of this function. He is also associated with the fundamental lemma of calculus of variations of which he proved a refined version based on that of Lagrange.[2][3]
- ^ Du Bois-Reymond, Paul (September 1875). "Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen". Mathematische Annalen (in German). 8 (3): 363–414. doi:10.1007/BF01443187. ISSN 0025-5831.
- ^ Dubois-Reymond: Erläuterungen zu den Anfangsgründen der Variationsrechnung. Mathematische Annalen, Band 15, 1879, S. 283–314, hier S. 297, 300.
- ^ Oskar Bolza: Vorlesungen über Variationsrechnung. Teubner 1909, S. 26. Nach Bolza stammt der älteste Beweis von Friedrich Stegmann, Lehrbuch der Variationsrechnung, Kassel 1854, dort werden aber einschränkendere Annahmen gemacht.