VI.67 Charles-Jean de la Vallée Poussin
b. Louvain, Belgium, 1866; d. Brussels, 1962 Analytic number theory, analysis
De la Vallée Poussin graduated in engineering (1890) and mathematics (1891) from the Université Catholique de Louvain, where he went on to teach mathematical analysis from 1891 until 1951. His lectures formed the basis for his renowned Cours d’Analyse Infinitésimale, which ran to many editions from 1903 to 1959. A member of the most famous academies in Europe and the United States, with honorary doctorates from Paris, Strasbourg, Toronto, and Oslo, he was the first president (1920) of the International Union of Mathematicians (now the International Mathematical Union). He was made a baron in 1930.
De la Vallée Poussin’s main achievement was his proof in 1896 of the PRIME NUMBER THEOREM [V.26] (an asymptotic estimate for the distribution of prime numbers in the integers), first conjectured by GAUSS [VI.26] in around 1793. (The theorem was also proved independently by HADAMARD [VI.65] in the same year, also using complex function theory.) Shortly afterward, de la Vallée Poussin followed his proof with a sharper error term (1899), which he extended to prime numbers in an arithmetic progression.
When LEBESGUE [VI.72] first published his INTEGRAL [III.55] in 1902, de la Vallée Poussin immediately grasped its importance and, using an original approach, described it in the second edition of his Cours d’Analyse (1908). In addition, he introduced the concept of the characteristic function of a set (1915), and shortly afterward gave a decomposition theorem for the measure generated by a continuous function of bounded variation (1916).
Of particular importance for approximation theory and the summation of series is de la Vallée Poussin’s convolution integral (1908), for approximating periodic functions by trigonometric polynomials. His other significant results in this field include a lower bound for the error in the best approximation of a continuous function by a polynomial (1910), and a convergence test and a summation method for Fourier series (1918).
In 1911 de la Vallée Poussin was responsible for suggesting the Belgian Academy prize question that led to Jackson’s and Bernstein’s theorems on the order of the best approximation of a continuous function by polynomials. His existence and uniqueness theorem for the Chebyshev problem for an overdetermined system of linear equations (1911) was an important step in LINEAR PROGRAMMING [III.84]; his interpolation formula (1908) was fundamental for sampling theory; and his characterization of new classes of quasi-analytic functions by the rate of decrease of their Fourier coefficients (1915) was a notable development.
De la Vallée Poussin’s other achievements include determining a uniqueness condition for multipoint boundary-value problems (1929), which was a significant result for the study of nonoscillatory solutions of linear differential equations; and solving various problems of the conformal representation of multiply connected regions (1930-31). In potential theory he extended the concept of capacity to arbitrary bounded sets, proved his extraction theorem for bounded sequences of set functions, and, by introducing measure theory into POINCARÉ’S [VI.61] “méthode de balayage” (“sweeping-out method”) for the DIRICHLET PROBLEM [IV.12 §1], he paved the way for modern abstract potential theory.
Further Reading
Butzer, P., J. Mawhin, and P. Vetro, eds. 2000-4. Charles-Jean de la Vallée Poussin. Collected Works—Oeuvres Scientifiques, four volumes. Bruxelles/Palermo: Académie Royale de Belgique/Circolo Matematico di Palermo.