Rachel Schoenberg was fluent as writing and speaking French and she taught Isaac the language. This enable him to read French physics books from the age of twelve when he became interested in the topic. However, to understand physics required mathematics and very soon Isaac had become more interested in mathematics than in physics. He read many mathematics textbooks in French, including Jacques Hadamard's Leçons de Géométrie Elémentaire Ⓣ and other mathematics books in the same series. He studied at the High School in Iasi but during 1917-18 and 1918-19 he studied at home for his Baccalaureate examinations. Titu Patriciu, a mathematics teacher at the High School, helped him with his mathematical studies. After sitting his Baccalaureate examinations in 1919, later in the same year he began his studies at the University of Iasi.
In his first year, 1919-20, Schoenberg studied Victor Costin's course in 'Projective and Descriptive Geometry' and, in 1920-22, he studied the following courses leading to the degree of M.A.: 'Analytic Geometry', taught by Alexandru Myller; 'Algebra Including Galois Theory', taught by Simion Stoilow; 'Analysis', taught by Simion Sanielevic; 'The Theory of Analytic Functions', taught by Vera Myller-Lebedev; 'Mechanics', taught by Simion Sanielevici; and 'Astronomy', taught by Constantin Popovici. He also took some extra courses which were not part of the M.A. syllabus: 'Differential Geometry', taught by Alexandru Myller; and 'Differential Equations including Fuchs Theory', taught by Simion Sanielevici. Let us say a little about some of those who taught Schoenberg at the University of Iasi. Alexandru Myller (1879-1965) studied with Felix Klein and David Hilbert at Göttingen from 1902 and was awarded his Ph.D. in 1906. He was appointed in 1910 as a professor at the Department of Analytical Geometry at the University of Iasi. Simion Sanielevici (1870-1963) studied at the University of Bucharest and then undertook research in Paris. He was awarded a doctorate by the Sorbonne in 1909. After teaching in Bucharest he was appointed as a professor at the University of Iasi in 1920. Vera Myller-Lebedev (1880-1970), born Vera Myller, studied at St Petersburg and then undertook research at Göttingen where she met Alexandru Myller; they later married. She was awarded a Ph.D. from Göttingen in 1906 and worked at the University of Iasi from 1911. She became a professor in the department of higher algebra and functions theory in Iasi in 1918. Taught by these excellent mathematicians, Schoenberg received his M.A. in Mathematics (with distinction) in 1922.
During 1922-25 he studied at Germany. The first semester of 1922-23 he spent in Göttingen, then he spent the second semester of that academic year at Berlin. There were two main attractions to Berlin; his sister Irma was there studying piano and he was enthused by Issai Schur's courses at the University of Berlin. He remained in Berlin for the whole of the academic year 1923-24 and during the three semesters there he attended Schur's courses on 'Algebra', 'Number Theory', and 'Analytic Number Theory'. It was the 'Analytic Number Theory' course, in which Schur had discussed Hermann Weyl's results on the uniform distribution of numbers mod 1, that gave Schoenberg the ideas which would eventually form his Ph.D. thesis. Göttingen was also very attractive since Edmund Landau was teaching there, so Schoenberg spent the whole of the academic year 1924-25 at Göttingen. He attended Edmund Landau's courses on 'Entire Functions', 'Trigonometric Series', 'The Big Fermat Problem', and 'Analytic Number Theory'. By the time he returned to Göttingen in 1924, Alexander Ostrowski was teaching there and Schoenberg attended his course on 'Overconvergence of Power Series' and also the seminar that Ostrowski ran.
Schoenberg had obtained his Baccalaureate in Romania and this was not recognised by Germany so he was not allowed to submit his thesis to a German university. He therefore returned to the University of Iasi in 1925 where he was appointed as an assistant to Victor Costin but his thesis advisor was now Simion Sanielevici. Schoenberg presented his thesis Über die asymptotische Verteilung reeller Zahlen mod 1 Ⓣ to the University of Iasi and was awarded his Ph.D. in June 1926. All High School graduates had to serve for a year in the army so Schoenberg spent six months at the Field Artillery School in Timisoara, a city that had been allotted by the Treaty of Trianon to Romania in 1920. Graduating as a corporal, he spent the following six months in a horse drawn Field Artillery Regiment near Chisinau in Bessarabia. Chisinau was a city which was Russian before it was allotted by the Treaty of Trianon to Romania in 1920. Schoenberg writes :-
The year 1926-1927 was as peaceful as Europe experienced between the wars. In recollection and by old photographs my military service has some elements of musical comedy. As a lifelong vegetarian I could not eat at the officers' mess, but with my doctorate in Mathematics I was well treated.At Göttingen Schoenberg had got to know Edmund Landau well and it was Landau who arranged a visit for Schoenberg to the Hebrew University of Jerusalem which he made in the spring semester of 1928. Edmund Landau, an enthusiastic Zionist, was one of founders of the Hebrew University of Jerusalem in 1925 and worked there until 1930. Schoenberg lectured in Hebrew in Jerusalem on Higher Algebra. It was during this visit that :-
... Schoenberg became interested in estimating the number of real zeros of a polynomial and so began his very influential work on Total Positivity and Variation diminishing linear transformations...Landau, however, proved important in other ways in Schoenberg's life for in Jerusalem he met Landau's elder daughter. In 1930, after his return from Jerusalem, Schoenberg married Landau's daughter Charlotte in Berlin. Charlotte, known as Dolli, had been born on 12 March 1907 in Berlin. This was not Schoenberg's only mathematical connection by marriage since his sister Irma, a concert pianist, married Hans Rademacher.
In 1930 Schoenberg was awarded a Rockefeller fellowship which enabled him to go with his wife to the United States. There he was a postdoctoral worker at Chicago during 1930-31, where he studied the Calculus of Variations and collaborated with Gilbert Bliss. This went very well and Bliss arranged for Schoenberg to remain at Chicago for the year 1931-32 as his assistant. During these two years Schoenberg wrote seven papers, two of which were jointly written with Bliss. These papers are: (with Gilbert Bliss) On separation, comparison, and oscillation theorems for self-adjoint systems of linear second order differential equations (1931); The minimizing properties of geodesic arcs with conjugate end points (1931); On finite and infinite completely monotonic sequences (1932); On finite-rowed systems of linear inequalities in infinitely many variables I (1932); On finite-rowed systems of linear inequalities in infinitely many variables II (1932); (with Gilbert Bliss) On the derivation of necessary conditions for the problem of Bolza (1932); and Some applications of the calculus of variations to Riemannian geometry (1932). While in Chicago, Iso and Dolli Schoenberg's first child Elizabeth was born on 17 September 1931.
In 1932 the family left Chicago and went to Cambridge Massachusetts where he attended courses both at Harvard and at the Massachusetts Institute of Technology. At Harvard he attended lectures by David Vernon Widder (1898-1990), and at M.I.T. he studied the courses given by Dirk Struik and Jesse Douglas. In 1933 he became a member of the Institute for Advanced Study at Princeton and he remained a member until 1935. During this time he received some financial support working for the Annals of Mathematics. At Princeton, influenced by Leonard Mascot Blumenthal (1901-1984) who was a visiting scholar at the Institute for Advanced Study from 1933 to 1936, he began working on distance geometry. His work on the Annals of Mathematics led to him reading a paper by Maurice Fréchet which motivated him to write nine papers on :-
...the isometric imbedding of metric spaces into Hilbert space and positive definite functions.Schoenberg was acting assistant professor at Swarthmore College from January 1935 until June 1936. He was then appointed to Colby College in Waterville where he spent the next five years. The article  gives an excellent picture of his work at Colby in 1940. He is described as:-
... a man in his late thirties, whose dark hair, ruddy high cheekbones and flashing grin give his maturity a boyish expression. ... Strangely enough, it is the elementary classes which he particularly enjoys teaching. He says that it keeps him conscious of fundamentals and he enjoys the challenge of making students see the fascination of his subject. Something of Schoenberg's approach is seen in a new course called "Non-technical mathematics" which he introduced last year. Aimed for students who do not plan to take any further work in mathematics, he is freed from the necessity of drilling them on fundamentals preparatory for advanced work, and so can teach the subject solely for its cultural and mind-stretching values. Students say it is a hard course, but fun. ... His classroom manner is vigorous, energetic, never losing the attention of the class, and using the blackboard constantly and effectively. One notes his vivid and concrete analogies, his ability to crystallise abstract concepts. English is, of course, an acquired language, and yet he has an extraordinary gift for precision of phrase, and he talks with wit and grace. It is more than linguistic ability, it reveals clarity of mind and a quality of personal distinction.Perhaps, given these comments about his ability in the English language, it is a good place to note that the Schoenberg archive  contains letters, notes, and documents, written in Romanian, German, French, Italian, Dutch, and Russian. On 17 October 1937 their second daughter, Beatrice, was born in Waterville. Five months later Schoenberg's father-in-law Edmund Landau died and his mother-in-law Marianne Landau (1886-1963), a daughter of Paul Ehrlich (Nobel Laureate and founder of Chemotherapy), emigrated to the USA and joined them in Waterville.
In 1941, he was appointed to the faculty at the University of Pennsylvania. During the years 1943-45 he was released from the University of Pennsylvania for war work as a mathematician at the Army's Ballistic Research Laboratory in Aberdeen, Maryland (the Aberdeen Proving Ground). It was during this war work that he initiated the area for which he is most famous, the theory of splines. Karlin writes in :-
... Schoenberg is noted worldwide for his realisation of the importance of spline functions for general mathematical analysis and in approximation theory, their key relevance in numerical procedures for solving differential equations with initial and/or boundary conditions, and their role in the solution of a whole host of variational problems. The fundamental papers by Schoenberg [two papers in 1946] form a monument in the history of the subject as well as its inauguration.The authors of  state:-
For the next 15 years, Schoenberg had splines all to himself. This changed around 1960, when computers became more widespread and splines first assumed their role as the premier tool for data fitting and computer-aided geometric design. Schoenberg's more than 40 papers on splines after 1960 gave much impetus to the rapid development of the field.Although his research was producing extremely important mathematics, he suffered a family tragedy :-
My family received a heavy blow when my beloved wife Dolli died of acute Leukemia on July 2, 1949. My older daughter Elizabeth left for College (Radcliffe) in the Fall of the same year. On December 2, 1950, I married Dolly van der Hoop, of Amsterdam, who was visiting Mark Dresden and his wife in Media PA.Iso and Dolly Schoenberg had a son Michael Jan, born in Santa Monica, California, on 12 September 1951.
Schoenberg made further outstanding contributions in a series of papers between 1950 and 1959 on the theory of Pólya frequency functions. His work here extended that begun by Pólya, Laguerre and Schur on approximating functions by polynomials with only real zeros. This work led Schoenberg to discover remarkable properties of polynomials all of whose zeros are negative and real.
In 1966 Schoenberg moved from the University of Pennsylvania to the University of Wisconsin where he became a member of the Mathematics Research Center. He remained at Wisconsin-Madison until he retired in 1973. However, he continued to produce important works after he retired and of his 174 papers and books, over 50 appeared after his retirement. During his time at Wisconsin, Schoenberg introduced another concept of major importance, namely cardinal splines. He investigated their wide applications in approximation theory in a series of three papers between 1969 and 1973.
Schoenberg published joint papers with a number of mathematicians including his brother-in-law Rademacher. He also collaborated with Besicovitch, Erdős, Curry, von Neumann and Szegő. Although he never produced a joint publication with his father-in-law Edmund Landau, he did spend a great deal of his time working on problems that Landau had considered.
In , written at the time he retired in 1973, his interests were described:-
Schoenberg is a man of broad culture, fluent in several languages, addicted to art, music and world literature, sensitive, gracious and giving in all ways. [He] frequently builds physical models related to his mathematical enquiries. ... The working desk at his home where he engages in research is actually a draftsman's bench complete with T-square, etc. and a tall stool. Mobiles, artistic works, models of ruled surfaces, icosahedrons and other objects are strewn throughout the room. English, French and German novels, numerous paintings and artefacts are scattered on all the nearby easy chairs. He buys and collects books of all vintages with passion. Historical mathematical discourses especially fascinate him and his articles frequently reflect this interest.We noted above that retirement did not slow Schoenberg's productivity. In 1983 his book Mathematical time exposures was published by the Mathematical Association of America. You can read extracts from a review of this book at THIS LINK.
Article by: J J O'Connor and E F Robertson
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