Baron Charles Jean de la Vallée Poussin
Poussin, c. 1900
Born	Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin (1866-08-14)14 August 1866 Leuven, Belgium
Died	2 March 1962(1962-03-02) (aged 95) Watermael-Boitsfort, Brussels, Belgium
Citizenship	Belgium
Alma mater	Catholic University of Leuven (1834–1968)
Known for	Poussin graph Poussin summability Poussin theorem Prime number theorem
Awards	Poncelet Prize(1916)
Scientific career
Fields	Mathematics
Institutions	Catholic University of Leuven (1834–1968)
Doctoral students	Georges Lemaître

Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin (French pronunciation:[ʃaʁlʒɑ̃etjɛnɡystavnikɔlabaʁɔ̃dəlavalepusɛ̃]; 14 August 1866 – 2 March 1962) was a Belgianmathematician. He is best known for proving theprime number theorem.

The King of Belgium ennobled him with the title ofbaron.

De la Vallée Poussin was born inLeuven, Belgium. He studiedmathematics at theCatholic University of Leuven under his uncle Louis-Philippe Gilbert, after he had earned hisbachelor's degree inengineering. De la Vallée Poussin was encouraged to study for a doctorate inphysics and mathematics, and in 1891, at the age of just 25, he became anassistant professor in mathematical analysis.

De la Vallée Poussin became a professor at the same university (as was his father,Charles Louis de la Vallée Poussin, who taughtmineralogy andgeology) in 1892. De la Vallée Poussin was awarded with Gilbert's chair when Gilbert died. While he was a professor there, de la Vallée Poussin carried out research in mathematical analysis and the theory of numbers, and in 1905 was awarded the Decennial Prize for Pure Mathematics 1894–1903. He was awarded this prize a second time in 1924 for his work during 1914–23.

In 1898, de la Vallée Poussin was appointed as the correspondent to theRoyal Belgian Academy of Sciences, and he became a Member of the academy in 1908. In 1923, he became the President of the Division of Sciences.

In August 1914, de la Vallée Poussin escaped from Leuven at the time of its destruction by the invadingGerman Army ofWorld War I, and he was invited to teach atHarvard University in the United States. He accepted this invitation. In 1918, de la Vallée Poussin returned to Europe to accept professorships inParis at theCollège de France and atthe Sorbonne.

After the war was over, de la Vallée Poussin returned to Belgium, The International Union of Mathematicians was created, and he was invited to become its president. Between 1918 and 1925, de la Vallée Poussin traveled extensively, lecturing inGeneva,Strasbourg, andMadrid. and then in the United States where he gave lectures at the Universities of Chicago, California, Pennsylvania, and Brown University, Yale University, Princeton University, Columbia University, and the Rice Institute of Houston.

He was awarded thePrix Poncelet for 1916.^[1] De la Vallée Poussin was given the titles of Doctor Honoris Causa of the Universities of Paris, Toronto, Strasbourg, and Oslo, an Associate of the Institute of France, and a Member of thePontifical Academy of Sciences,^[2] Nazionale dei Lincei, Madrid, Naples, Boston. He was awarded the title of Baron byKing Albert I of the Belgians in 1928.

In 1961, de la Vallée Poussin fractured his shoulder, and this accident and its complications led to his death inWatermael-Boitsfort, nearBrussels, Belgium, a few months later.^[3]

A student of his,Georges Lemaître, was the first to propose theBig Bang theory of the formation of theUniverse.

Although his first mathematical interests were in analysis, he became suddenly famous as he proved theprime number theorem independently of his coevalJacques Hadamard in 1896.

Afterwards, he found interest inapproximation theory. He defined, for anycontinuous functionf on the standardinterval${\displaystyle [-1,1]}$, the sums

${\displaystyle V_n=\frac{S_n+S_{n+1}+\cdots +S_{2n-1}}{n}}$,

where

${\displaystyle S_n=\frac{1}{2}{c}_0(f)+\sum _{i=1}^n{c}_i(f)T_i}$

and

${\displaystyle c_i(f)}$

are the vectors of thedual basis with respect to thebasis ofChebyshev polynomials (defined as

${\displaystyle (T_0/2,T_1,\dots ,T_n).}$

Note that the formula is also valid with${\displaystyle S_n}$ being theFourier sum of a${\displaystyle 2\pi }$-periodic function${\displaystyle F}$ such that

${\displaystyle F(\theta )=f(\cos \theta ).}$

Finally, the de la Vallée Poussin sums can be evaluated in terms of the so-calledFejér sums (say${\displaystyle F_n}$)

${\displaystyle V_n=2F_{2n-1}-F_{n-1}.}$

The kernel is bounded (${\displaystyle V_n\le 3}$) and obeys the property

${\displaystyle f\ast V_n=f}$, if${\displaystyle f(x)=\sum _{j=-n}^n{a}_j{e}^{ijx}.}$

Later, he worked onpotential theory andcomplex analysis.

He also published a counterexample toAlfred Kempe's false proof of thefour color theorem. ThePoussin graph, the graph he used for this counterexample, is named after him.

The textbooks of his mathematical analysis course have been a reference for a long time and had some international influence.^[4]

The second edition (1909–1912) is remarkable for its introduction of the Lebesgue integral. It was in 1912, "the only textbook on analysis containing both Lebesgue integral and its application to Fourier series, and a general theory of approximation of functions by polynomials".^[4]

The third edition (1914) introduced the now classical definition ofdifferentiability due toOtto Stolz. The second volume of this third edition was burnt in thefire of Louvain during theGerman invasion.

The further editions were much more conservative, returning essentially to the first edition. Starting from the eighth edition, Fernand Simonart took over the revision and the publication of theCours d’analyse.

• Œuvres, vol. 1 (Biography and number theory), 2000 (eds. Mawhin, Butzer, Vetro), vols. 2 to 4 planned
• Cours d´Analyse, 2 vols., 1903, 1906 (7th edition 1938), Reprint of the 2nd edition 1912, 1914 by Jacques Gabay,ISBN 2-87647-227-9 (deals only with real analysis).^[5] Online:
  • Cours d'analyse infinitésimale, Tome I^[6]
  • Cours d'analyse infinitésimale, Tome II
• Integrals de Lebesgue, fonctions d´ensemble, classes de Baire,^[7] 2nd edition 1934, Reprint by Jacques Gabay,ISBN 2-87647-159-0
• Le potentiel logarithmique, balayage et representation conforme, Paris, Löwen 1949
• Recherches analytiques de la théorie des nombres premiers, Annales de la Societe Scientifique de Bruxelles vol. 20 B, 1896, pp. 183–256, 281–362, 363–397, vol. 21 B, pp. 351–368 (prime number theorem)
• Sur la fonction Zeta de Riemann et le nombre des nombres premiers inferieur a une limite donnée, Mémoires couronnés de l Academie de Belgique, vol.59, 1899, pp. 1–74
• Leçons sur l'approximation des fonctions d'une variable réelle Paris, Gauthier-Villars, 1919,^[8] 1952

• Poussin proof
• Remez algorithm
• La Vallée-Poussin

• Charles Jean de la Vallée Poussin at theMathematics Genealogy Project
• Biographie Universelle, byDidot.
• O'Connor, John J.;Robertson, Edmund F.,"Charles Jean de la Vallée Poussin",MacTutor History of Mathematics Archive,University of St Andrews

• 1866 births
• 1962 deaths
• 19th-century Belgian mathematicians
• 20th-century Belgian mathematicians
• Number theorists
• Harvard University Department of Mathematics faculty
• Academic staff of the University of Paris
• Catholic University of Leuven (1834–1968) alumni
• Belgian barons
• Foreign associates of the National Academy of Sciences
• Scientists from Leuven

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