Delsarte
# Jean Frédéric Auguste Delsarte

### Born: 19 Oct 1903 in Fourmies, France

Died: 28 Nov 1968 in Nancy, France

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**Jean Delsarte**'s father was the head of a textile factory in Fourmies but in 1914 the German armies advanced on the town and Jean left his home town with all the family, except his father, and fled to safety. Jean's father remained in Fourmies trying to save the remnants of his destroyed factory.

In 1922 Delsarte entered the Ecole Normale Supérieure in Paris, graduating in 1925. After completing his compulsory military service, he wrote a doctoral thesis during the one year in which he held a research fellowship. In 1928 he was awarded his doctorate and, in the same year, he was appointed to a post at the University of Nancy. In fact Delsarte was to remain on the staff at Nancy for the rest of his career. Among the senior posts he held there was the post of Dean of Science which he held during the years 1945-48.

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 [1], 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 in R has at each point x a value equal to its mean value on every sphere of centre x, the 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 finite set of values. This result is explained in *Lectures on Topics in Mean Periodic functions and the Two Radius Theorem* published in Bombay in 1961.
Delsarte worked independently of other mathematicians and had very few research students. His work is highly original but, because he was rather isolated, had less influence than would otherwise have been the case.

Although Delsarte remained at Nancy all his life he did lecture in many different universities from 1950 onwards, particularly ones in India, North America and South America.

**References (4 books/articles)**

JOC/EFR April 1997