In 1922 Delsarte entered the École Normale Supérieure in Paris. There he met with many students who would become his friends and play a major role in his life from that point on. In particular he met with mathematics student André Weil and the physicist Yves Rocard who were both in the same year as Delsarte. After one year they were joined by Henri Cartan, Jean Coulomb who studied mathematical physics, Paul Dubreil, René de Possel and the future philosopher of mathematics Jean Cavaillès. Several entering the École Normale in 1924 also joined the group of Delsarte's friends; these included Marcel Brelot, Jean Dieudonné and Charles Ehresmann. In the following year Claude Chevalley and Jean Leray entered the mathematics classes adding to the group of friends and future colleagues who combined their talents in a remarkable collaboration as Bourbaki in 1935. It was a collaboration in which Delsarte took a leading role and, looking at reports of their meetings, one is struck with the humour, the passion, and the self-belief that these young men of the École Normale displayed.
Delsarte graduated from the École Normale in 1925. After completing his compulsory military training, he wrote a doctoral thesis during the year in which he held a Thiers Foundation research fellowship. He worked during that year at the private mansion of the Foundation, undertaking research for his doctoral thesis and also working on his first two papers Sur les rotations dans l'espace fonctionnel and Étude de certaines équations intégrales qui généralisent celles de Fredholm which were published by the Academy of Science. In March 1928 he was awarded his doctorate for his thesis Les rotations fonctionnelles on the same topic as the first of the above two papers. However, in November of the previous year, he had been appointed as Chargé de cours at the Faculty of Science in Nancy. In fact Delsarte was to remain on the staff at Nancy for the rest of his career but his first promotion was to Maître de Conférences in October 1928. In the following year Delsarte married Thérèse Sutter, the daughter of a doctor. The two had been friends from childhood and they made their home at 4 rue de l'Oratoire in Nancy; there they had two daughters, Chantal and Micheline. In 1932 Delsarte's outstanding research was recognised with an invitation to address the International Congress of Mathematicians in Zürich. He lectured to the Congress on Le groupe des transformations conformes dans l'espace de Hilbert .
Not only did Delsarte teach some outstanding mathematics courses at Nancy but he taught a public course on astronomy from 1929, continuing to give this when at Nancy for the rest of his career. In 1931 he gave a course on linear groups of transformations on Hilbert space when invited to give the prestigious Peccot Foundation lectures at the College of France. At Nancy he developed a new course on differential equations in the academic year 1933-34 and in the following year, also at Nancy, he gave a course on Riemann spaces and relativity. In addition he lectured at the Henri Poincaré Institute in Paris on Hilbert spaces. It was during his regular visits to Paris during 1934-35 that Delsarte became heavily involved in the Bourbaki project to write a new analysis textbook which expanded into the remarkable Éléments de Mathématique . In October 1936 he was appointed professor of higher analysis at Nancy.
Delsarte cooperated with André Weil and Henri Cartan, both by this time lecturers in Strasbourg, in organising a joint seminar programme between Nancy and Strasbourg. He was able to bring many of his outstanding colleagues to Nancy; Paul Dubreil taught there between 1933 and 1937, Jean Leray was apointed part-time lecturer in applied mathematics in 1936 becoming a lecturer in 1937, and then professor of applied mathematics in 1938. Jean Dieudonné was appointed as a lecturer in Nancy, also in 1938.
There was another aspect of Delsarte's work during the 1930s which is quite unusual for a young mathematician, for he involved himself in several major administrative roles. He headed the Entrance Board for the French baccalaureat in Poland during the year 1928-29 during which time he met and became friends with Stanislas Zaremba. He served on the admissions board of the École Centrale in Paris during the 1930s and also was head of research at the Centre National de la Recherche Scientifique from July 1932 to October 1936. He received the Victor Noury Foundation prize from the Academy of Sciences in 1935 in recognition of his outstanding contributions. His involvement in both teaching and research led him to write three artices which were published in 1939 giving constructive criticism of the way scientific research and higher education were organised in France. These articles were L'organisation de la recherche scientifique, De l'enseignement supérieur en France et spécialement des Facultés des Sciences , and Sur un projet de réforme de l'enseignement supérieur .
By 1939 he had shown his talents at research, teaching and administration of mathematics and was having a major impact on mathematics in France with his involvement in a wide range of activities. World War II was to have a major impact on Delsarte's career, as of course it did for so many of his friends. Already trained as an artillery officer during his time at the École Normale Supérieure, Delsarte was put in charge of a unit in September 1939 which he commanded until August 1940. Under his command the 8th Battery was led across the Jura and the Alps, bringing them to Alsace where they saw action before he was demobilized in Nimes. Delsarte now went to the Faculty of Sciences at Grenoble to replace Jean Favard, the professor of mathematics, who had been captured by the Germans and was being held as a prisoner of war in Germany. The safest course for Delsarte would have been to remain in the comparative safety of Grenoble as long as possible, but he did not choose the easy option, returning instead in September 1941 to Nancy which was in a dangerous closed area.
While the war slowly ran its painful course, Delsarte continued to undertake duties for French mathematics both in Nancy and in Paris at the Centre National de la Recherche Scientifique where he served on a pure mathematics commission, and also as an examiner for entry to various Écoles. He was awarded the Vailland Prize by the Academy of Sciences in 1944 and in the same year he became head of mathematics at Nancy. He held this position from that time until his death. Among the other senior posts he held there after the war was the post of Dean of Science which he held during the years 1945-48.
Before the start of World War II, Delsarte had brought a number of leading mathematicians to Nancy. He repeated this after the war ended bringing Laurent Schwartz (1945-1952), Roger Godement (1946-1955), Jean-Pierre Serrre (1954), and Jacques-Louis Lions (1954-1964). In 1953 he created the Élie Cartan Institute at Nancy which, to a certain extent, was modelled on the Henri Poincaré Institute in Paris. Delsarte's involvement with Bourbaki, and the continual appearance of members of Bourbaki at the Institute, led to widespread belief that the Élie Cartan Institute was the Bourbaki group. In 1959 Delsarte put out a statement clarifying the position:-
The Bourbaki Group is completely distinct from the Élie Cartan Institute, which deals only with the administration of the Bourbaki Group. The members of the Élie Cartan Institute are not the members of the Bourbaki Group, but the intersection of these two sets is nonempty.Let us look at some of Delsarte's mathematical contributions. A review of  states:-
Delsarte is best known for his work on mean periodic functions and translation operators, but he was an analyst of great power and originality, which he applied to many problems ...He published a series of papers on this topic in 1934-35: Les fonctions moyenne-périodiques (1934); Application de la théorie des fonctions moyenne-périodiques à la résolution de certaines équations intégrales (1934); Application de la théorie des fonctions moyenne-périodiques à la résolution des équations de Fredholm-Nörlund (1935); and Les fonctions moyenne-périodiques (1935). He also later published Fonctions moyenne-périodiques sur un groupe abstrait (1937) and Théorie des fonctions moyenne-périodiques de deux variables (1960).
Delsarte worked in analysis extending work on series expansions due to Whittaker and Watson. He was greatly influenced by their text A Course of Modern Analysis and by Watson's Treatise on the Theory of Bessel Functions. Dieudonné, writing in , says:-
These works had convinced him that a good understanding of the formal properties of [series expansions of functions] was necessary to a fruitful study of their domains of definition and their mode of convergence. This was the course he followed with remarkable success, opening up new fields of research that are still far from having been thoroughly explored.One of the most surprising of Delsarte's results was a generalisation of a result due to Gauss. Gauss had shown that if a continuous function f on Rn has at each point x a value equal to its mean value on every sphere of centre x, then f is harmonic. Delsarte showed that f is harmonic under the weaker condition that f (x) is the mean value on two spheres centre x, radius a and b, provided a/b does not take one of a particular finite set of values. A proof of this result was first sketched in his paper Note sur une propriété nouvelle des fonctions harmoniques (1958) and is explained in more detail in Lectures on Topics in Mean Periodic functions and the Two Radius Theorem published in Bombay in 1961.
Other important work by Delsarte was on transmutation operators. For example he published Hypergroupes et opérateurs de permutation et de transmutation (1956) and two papers with Jacques-Louis Lions both titled Transmutations d'opérateurs différentiels dans le domaine complexe and published in 1957. Levitan looks at this aspect of Delsarte's work in detail in .
Although Delsarte remained permanently on the staff at Nancy all his life, he did lecture in many different universities from 1947 onwards, particularly ones in India, North America and South America. He spent four months at the Institute for Advanced Studies at Princeton in the first half of 1947, then in each of the years 1948 to 1951 he spent four months each year at São Paulo in Brazil. He spent three months in Mexico in 1952, then three months in the United States and Canada in 1957. Two years later he was back in Mexico for three months and then in 1962-65 he spent three years in Japan before returning via China and Russia. In fact Delsarte had originally intended to spend four years in Japan but cut short his visit, partly due to health problems, in particular his eyesight was rapidly deteriorating. Back in Nancy he did not let his poor eyesight prevent him from attending several European conferences visiting Cambridge, Liège, Brussels, Louvain, Lausanne, Basel, and Zürich.
Despite his health problems, Delsarte taught the mechanics course in 1967-68. However, 1968 was a year of student unrest in France, and the students in Nancy looked for educational reforms. Delsarte had, of course, always advocated educational reforms, but he did not support the extreme methods adopted by some students. However, he wrote on 20 May 1968:-
The attitude of the students here is rather correct ... There is not complete occupation of the Faculty, since the offices of the Professors and the library are free.Near the end of September 1968 Delsarte suffered a heart attack. He wrote to the Faculty:-
I must tell you that I am currently a little ill: last Thursday I had, in my office in the Faculty, a rather spectacular heart attack, which I found extremely painful. For the moment I am in bed, and this evening the doctor will give me an electrocardiogram. I will then know the duration of my disability. I think that it will not exceed ten days, because I now feel very well.He died after suffering a second heart attack on 28 November.
Delsarte received many honours and we have already mentioned some above. After World War II he was awarded a medal from the University of Liège in 1950 and another from the University of Mexico in 1952. He was made Chevalier de la Légion d'Honneur in 1954, Commander of the Ordre des Palmes Académique in 1962, and received the Bordin Prize from the Academy of Sciences in 1964.
Article by: J J O'Connor and E F Robertson