# Arnaud Denjoy

Arnaud Denjoy ( born January 5, 1884 in Also in the Gers department, † January 21, 1974 in Paris) was a French mathematician who worked mainly in the real analysis.

## Life and work

Denjoy was the son of a wine merchant in Perpignan and a Spanish mother. He attended the schools of Auch and Montpellier, and studied from 1902 at the École Normale Supérieure with Emile Picard, Émile Borel and Paul Painleve. Denjoy was so good that he won a fellowship from the Thiers Foundation as a student. Under the influence of Émile Borel ( and the writings of Rene Baire ) he turned not to the real analysis. After the graduation in 1909 he went as a tutor ( Maitre of conferences ) at the University of Montpellier, a post previously held Rene Baire. In World War I he could not afford because of poor eyesight active military service in 1917 and took a professorship in Utrecht, where Johannes van der Corput was his assistant. From 1922 until his retirement in 1955 he was professor at the Sorbonne in Paris ( nominally on the occupied previously by Henri Poincaré Chair of celestial mechanics ).

In a series of works from the 1920s, he examined the calculation of the coefficients in convergent trigonometric series, summarized in a four-volume monograph which appeared from 1941 to 1949. It is also one of his most famous discoveries, the Denjoy integral, contain a generalization of the Riemann integral, today with the theory of the named partly after Henstock, Kurzweil or Perron integral ( also called its " Gauge Integral" ) merged. Further work of Denjoy related quasi- analytic functions. There, the theorem of Denjoy and Carleman is named after him, which specifies criteria that an analytic function is quasi- analytic ( Denjoy 1921). Important contributions made Denjoy also to the theory of dynamical systems, in particular to differential equations on the torus ( Poincaré - Denjoy theory ). The set of Denjoy (1932, Journal de Mathematiques ) gives criteria for when a diffeomorphism of a circle is conjugate to a rotation. 1931 ( Compte Rendus ) gave Denjoy a probabilistic interpretation of the Riemann Hypothesis.

Denjoy was also a friend of Nikolai Nikolaevich Luzin the Russian mathematician and had contacts with the mathematicians of the school.

Denjoy was also politically active. He supported the Radical Party of multiple French President of the Council Édouard Herriot and was for this city council in 1912 from Montpellier and from 1920 Chief Executive of the Department of Gers.

In 1941 he was elected to the French Academy of Sciences, which he was president in 1962 ( he was also a member of the Academies of Amsterdam, Warsaw and Liège ). In 1954 he was vice-president of the International Mathematical Union. In Russia ( where he stood with Luzin in correspondence) 1970 he was awarded the Lomonosov Gold Medal. He was a member of the Russian Academy of Sciences since 1971. In 1962 he was invited speaker at the International Congress of Mathematicians in Stockholm (Les differential equations Periodiques ).

He was married in 1923 and had 3 sons.

The asteroid ( 19349 ) Denjoy was named after him.