From his boyhood he was encouraged by the mathematician Louis-Philippe Gilbert but at first Vallée Poussin thought he would become a Jesuit priest. He entered the Jesuit College at Mons but he found the teaching there unacceptable. He was particularly disappointed in the teaching of philosophy at the College, so he turned to a different topic although he still did not have mathematics as his main interest. He studied engineering and obtained his diploma in that subject. However soon after this he became absorbed by pure mathematics. He studied at the University of Louvain where he was taught by Gilbert who proved to be an inspiring teacher. Gilbert was an excellent mathematician and the author of a fine analysis textbook. Vallée Poussin also studied at the University of Paris and at the University of Berlin.
In 1891 Vallée Poussin was appointed as an assistant of Gilbert's at the University of Louvain. However the collaboration was not to last for long since Gilbert died in 1892. Although only 26 years old at the time Vallée Poussin was elected to Gilbert's chair.
Vallée Poussin's first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations. One of his first papers in 1892 on differential equations was awarded a prize by the Belgium Academy. His best known work, however, appeared four years later in 1896 when he proved the prime number theorem. This states that π(x), the number of primes ≤ x, tends to x/logex as x tends to infinity.
The prime number theorem had been conjectured in the 18th century, but in 1896 two mathematicians independently proved the result, namely Hadamard and Vallée Poussin. The first major contribution to proving the result was made by Chebyshev in 1848, then the proof was outlined by Riemann in 1851. The clue to two independent proofs being produced at the same time is that the necessary tools in complex analysis had not been developed until that time. In fact the solution of this major open problem was one of the major motivations for the development of complex analysis during the period from 1851 to 1896.
In 1900, while on holiday in Norway, Vallée Poussin met a Belgium family. He married the talented daughter of this family and it was a very happy marriage. The result was that he had a home where he and his wife were happy and contented. He lived in Louvain from the time he was first appointed there except for a few periods abroad. During the First World War he was invited to Harvard in 1915 and then to Paris in 1916. Among a number of famous lectures he gave were those in Fribourg in 1918, Rome in 1923 and Houston in 1924.
Other than the prime number theorem, Vallée Poussin's only contributions to prime numbers were contained in two papers on the Riemann zeta function which he published in 1916. The Riemann hypothesis, perhaps the most famous of all the still open questions of mathematics, is that all the complex zeros of the zeta function lie on the line 1/2 + i b. Vallée Poussin strengthened results proved by Hardy in 1914 which showed that an infinite number of the zeros were on that line. Vallée Poussin's results were of passing interest, however, for Hardy and Littlewood proved still stronger results in 1918.
Vallée Poussin also worked on approximation to functions by algebraic and trigonometric polynomials from 1908 to 1918. Let us quote Vallée Poussin's own description of the problem of approximation as given in a lecture which he gave in Houston in 1924:-
The most important of the problems which have been attacked in the study of approximation is that of the order of approximation. Let us define first what we mean by approximation. For example, let a continuous function f (x) be represented by means of a polynomial of degree n, and let Pn(x) be such a polynomial. The difference f - Pn is the error of the approximation, and is a function of x; its maximum value in the interval of representation is the approximation on. This positive number approaches 0 as 1/n approaches zero, if the polynomial Pn is well chosen. ... The problem of the order of approximation is the following: To determine the relation which exists between the order of approximation on, which f (x) may admit for a finite expression of order n, and the differential properties of the function.He then continued to put his own contribution to this problem into context, although one must say that it is phrased in a very modest way:-
I offered myself the beginnings of an answer to this very problem in 1908, while studying the approximation given by Edmund Landau's integral. I showed also that the function |x| admits an approximation of the order of 1/n by a polynomial of degree n, and I raised the question of deciding whether or not that was the order of the best possible approximation. This definite question had much more importance for the development of the subject than the few isolated results which I had obtained, because that question caused the writing of the two most important memoirs on the subject, one by D Jackson and the other by Sergei Bernstein.Vallée Poussin's most major work was Cours d'analyse . Burkill writes in :-
It was [Jordan's Cours d'Analyse] which, as is recorded by Hardy and other mathematicians of his generation, opened their eyes to what analysis really was. If Jordan's is the most noble of the Cours d'Analyse and perhaps Goursat's (helped by its translation by Hedrick) the most widely read, it can hardly be doubted that Vallée Poussin's is the most elegant and lucid.Vallée Poussin's Cours d'analyse Ⓣ went through several editions, each containing new material. By 1899, several years before the publication of the first edition, much of the material already existed in the form of lecture notes. The first edition of Volume 1 appeared in 1903, and the first edition of Volume 2 in 1906. Volume I covered differentiation of functions of one or more variables, and integration of functions of a single variable. Volume 2 covered multiple integrals, differential equations, and differential geometry. The treatise was written in an interesting way, combining an introductory text with an advanced work for specialists. The way this was achieved was having two different type sizes. If a reader only read the larger type then it was a complete introduction to the subject for beginners or those interested in applications to engineering. The smaller type material was aimed at the pure mathematical specialist interested in the deeper subtleties.
The work changed dramatically when a second edition appeared, Volume 1 in 1909 and Volume 2 in 1912. Most of the additional material appeared in small type and covered topics such as set theory, in particular the Schröder-Bernstein theorem, the Lebesgue integral, functions of bounded variation, the Jordan curve theorem, polynomial approximation, Parseval's theorem on trigonometric series, results of Fejér, etc.
The third edition of Volume I again contained new material and was published in 1914. However World War II disrupted Vallée Poussin's work. The promised German translation failed to appear and the third edition of Volume 2 was burned by the German army when it overran Louvain. It would have discussed the Lebesgue integral, work which was never to be published in this form but a lot of it was incorporated into a later monograph. Unlike many similar books of its time Cours d'analyse Ⓣ contains no complex function theory. The fourth edition appeared in 1921 and 1922. It ended the larger/smaller print distinction and became a work aimed at beginners. The two volumes had reached their seventh edition by 1938 but it went through much fewer changes after the fourth edition.
After 1925 Vallée Poussin turned to complex variable, potential theory and conformal representation. Further important texts published by him were his Borel tract on the Lebesgue integral (1916), approximation theory (1919), mechanics (1924), and potential theory (1937). In 1930 Vallée Poussin was revising his 1916 tract Lebesgue integrals: Set functions: Baire classes when Luzin's Lectures on analytic sets and their applications was published. The paper  contains three letters written by Vallée Poussin to Luzin dated 4 February 1933, 8 March 1933 and 21 March 1933. Vallée Poussin comments in these letters on the fact, which is of great interest to him, that Luzin used slightly different classifications of the same sets as he had studied. He gives high praise to Luzin's book.
Publication of Vallée Poussin's work Le potential logarithmique Ⓣ was held up by World War II and only published in 1949.
Vallée Poussin was elected to the Belgium Academy in 1909. More honours were to follow including election to the Madrid Academy of Sciences, the Naples Society of Science, the American Academy of Arts and Sciences, the Institute of France, the Accademia dei Lincei, the Paris Academy of Science, and the American National Academy of Sciences. There were celebrations in 1928 when Vallée Poussin had held the chair at the University of Louvain for 35 years and again celebrations in 1943 when he had been 50 years in the chair of mathematics at Louvain.
In 1928, when he had held the chair at Louvain for 35 years, the King of Belgium conferred the title Baron on Vallée Poussin at the celebrations for this event. In 1961 he fractured his shoulder and since Vallée Poussin was in his mid 90s it failed to heal. His death followed a few months later.
Article by: J J O'Connor and E F Robertson
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