After being awarded a Master's degree by the University of Wisconsin in 1917 he returned to Harvard where he was appointed as an instructor in mathematics. He registered for doctoral studies and began research under the supervision of Maxime Bôcher. By this time the United States had entered World War I and Walsh enlisted in the U.S. Navy. He served on troop transport vessels in the North Atlantic until he returned to his position in Harvard in 1919. By this time Bôcher had died so Walsh had to begin working under a new research supervisor. He now worked with G D Birkhoff who himself had been inspired by Bôcher. Walsh was awarded his doctorate by Harvard in 1920 for his thesis On the location of the roots of a Jacobian of two binary forms, and of the derivative of a rational function. He had published a seven page paper with this title in the Transactions of the American Mathematical Society in 1918 but even this was not his first publication for in 1916 he had published Note on Cauchy's integral formula in the Annals of Mathematics.
A Sheldon Travelling Fellowship allowed Walsh to spend the academic year 1920-21 in Paris where he worked with Montel. He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921. Back at Harvard he was appointed as an instructor in 1921, then three years later was promoted to assistant professor. One publication in 1923 which was not seen to be particularly significant at the time was on orthogonal expansions. Later the functions studied in this work were called Walsh functions and they proved an important tool in digital and signal processing. It is Walsh functions that have had the biggest impact of any of Walsh's work but nobody would have guessed this in 1923, least of all Walsh. Rising steadily through the ranks at Harvard he was promoted to associate professor in 1930, then full professor in 1935. He was chairman of the Mathematics Department from 1937 to 1942. During this 1921-1942 Harvard period, he had spent a year (1925-26) at the University of Munich where he worked with Carathéodory, and a year at the Institute for Advanced Study at Princeton (1934-35). He married Aline Natalie Burgess on 15 July 1931 but the marriage ended in divorce and he married for the second time on 2 April 1946 to Elizabeth Cheney Strayhorn who he met when she was an officer in the WAAC; he had two children with his second wife.
Walsh had a remarkable publication record. The obituary  by Morris Marden (a student of Walsh) lists 279 articles, 7 books and 31 PhD students. He studied the relative location of the zeros of pairs of rational functions, zeros and topology of extremal polynomials, the critical points and level lines of Green's functions and other harmonic functions, conformal mappings, Padé approximation, and the interpolation and approximation of continuous, analytic or harmonic functions. Sewell writes in :-
Polynomial approximation was neither discovered nor invented by J L Walsh (which may come as a surprise to some mathematicians). He is the one individual, however, who took a few scattered results on the subject and extended them, added mightily to them, and knit the whole together into a comprehensive, coherent theory.His books include Interpolation and approximation by rational functions in the complex domain (1935), Approximation to polynomials in the complex domain (1935), A Bibliography on orthogonal polynomials (1940), The location of critical points of analytic and harmonic functions (1950), Approximation by bounded analytic functions (1960), and The theory of splines and their applications (1967).
During World War II Walsh again took up his naval service. He was Lieutenant Commander from 1942 to 1943, then Commander from 1944 to 1946. After the war ended he retained his connection with the navy as a member of the Naval Reserve. He was a Captain for several years in the 1950s before he retired from the service in 1955. He returned to Harvard after his war service in 1946 when he was named Perkins Professor of Mathematics. He held this position for 20 years until he retired in 1966.
As a lecturer Walsh believed in teaching a wide variety of courses. The topics he taught, rotating them from year to year, included calculus, algebra, mechanics, differential equations, complex variable, probability, number theory, potential theory, approximation theory, and function theory.
Walsh was honoured with election to the National Academy of Sciences (United States) in 1936 and served as vice president of the American Mathematical Society in 1937 and president in 1949-50. After retiring from Harvard he moved to the University of Maryland choosing his home in College Park which :-
... had to be at the right walking distance from the University.He continued to work with research students and postdoctoral students at Maryland up to a few months before his death. He died at his home.
Sewell writes :-
... an outstanding ... characteristic of his professional career is his unselfish promotion of research. He has always encouraged others to attack problems which he has proposed ... Throughout his books and articles you will find open problems and he has worked with his students, former students, and colleagues frequently more to encourage them than to produce another paper under his own name.This same sentiment is expressed by Widder in :-
Surely the outstanding attribute of this man, that would occur first to any of his colleagues, was his devotion to research. He neglected none of his duties connected with his job, but he always arranged that they left him time for his writing. Even as departmental chairman or as president of the American Mathematical Society he remained faithful first and foremost to research.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page