After graduating from the Gymnasium, Fekete entered the University of Budapest to study mathematics. He was taught by Lipót Fejér and greatly influenced by him - this is hardy surprising, since Fejér inspired a whole generation of Hungarian mathematicians. Just a little note at this point; in Hungarian 'Fekete' means 'black' and 'Fejér' means 'white' so the student and his advisor were 'black and white'! Fekete was awarded a doctorate by the University of Budapest in 1909 but already he had a number of papers related to his doctoral thesis in print: A general treatment of linear congruence systems (Hungarian) (1908), Über die additive Darstellung einiger zahlentheoretischer Funktionen Ⓣ (1908), and On the additive representation of some number theoretical functions (Hungarian) (1909). After the award of his doctorate, Fekete went to Göttingen to undertake postdoctoral studies in 1909-10 at the Georg-August University of Göttingen. There he worked with Edmund Landau and during his year in Göttingen wrote a number of further papers including: On necessary and sufficient conditions for the summability of power series (Hungarian) (1910), Sur les séries de Dirichlet Ⓣ (1910), and Sur un théorème de M Landau Ⓣ (1910). However, it was Fejér, not Edmund Landau, who was the greatest influence on Fekete as Rogosinski explains :-
Fekete as a mathematician was very typically an analyst of the school of L Fejér. From him he inherited the delight in a particular isolated problem. From him too he learned the elegant simplicity of his analytical technique and style. Very little influence of his second teacher Edmund Landau is seen in this.After his year in Germany, Fekete returned to Budapest where he taught in secondary schools for eighteen years. He met the mathematics teacher Dora Lenk and, in 1914, they married. Michael and Dora Fekete had two sons but, sadly, Dora died in 1922. During these years in which he was a school teacher, Fekete also taught at the university as a docent. One of the university students he taught was John von Neumann and they published the joint paper Über die Lage der Nullstellen gewisser Minimum Polynome Ⓣ in 1922. In fact Fekete had taught von Neumann while he was still at school as he had been employed as a private tutor. By this time Fekete had already published about 20 papers, but this was von Neumann's first paper published only one year after he completed his studies at the Gymnasium. This paper looked at the transfinite diameter of a set, a concept which Fekete worked on throughout the rest of his career.
Fekete emigrated in 1928 when he became a lecturer at the Hebrew University of Jerusalem. It was a new university, founded on Mount Scopus three years before Fekete took up the lectureship there. After working as a lecturer for a year, he was made a professor and appointed as Director of the University's Einstein Institute of Mathematics :-
At the university he played an important role in the administration, was Dean of Science, and later the Rector from 1945-1948.His period as rector was important for the Hebrew University for in 1948 Jerusalem was divided into Israeli and Jordanian sectors with Mount Scopus in the east part which became Jordanian. The university was then moved to Giv'at Ram in the Israeli part. Fekete retired in 1955 and in the year of his retirement he attended the International Congress of Mathematicians in Amsterdam. He gave his lecture Transfinite diameter and Fourier series to the congress on Monday, 6 September, to Section IId with Jean Leray in the chair. Also in the year in which he retired he received his greatest honour when presented with the Israel Prize for Exact Sciences.
Retirement did not mean that Fekete gave up research but continued working on ideas that had fascinated him throughout his career. In 1958, one year after his death, his paper New methods of summability was published by the London Mathematical Society. V F Cowling begins a review (which explains how it came to be written) as follows:-
In 1916 the author showed that analytic continuation of a power series can be represented as a matrix-transformation of the partial sums by [a specific] upper triangular matrix .... This appeared in a Hungarian textbook by Beke (1916) an English summary of which is to be found in a paper of Vermes (1949). This method of summability has subsequently been called the 'Taylor method'. In the present note (a summary of a lecture given by the author in 1954 as compiled by P Vermes) two new methods of summability are introduced.In  Fekete's last few years are described; note that he married Erna Baruch during these late years:-
Fekete was a genuine and enthusiastic mathematician and a very fertile one, as can be seen from the long list of his publications [containing 77 items]. Even as an old man he had preserved his youthful enthusiasm and his capacity for work - in fact, he died over his desk doing mathematics. He travelled widely in his late years, both in Europe and in the United States of America, and loved lecturing on his problems wherever he could. The small energetic man with fiery eyes and an unruly Einstein mane of white hair on his fine large head was a well-known visitor and speaker at mathematical conferences and seminars everywhere.Fekete advised several doctoral students who went on the become world-leading mathematicians, perhaps the most famous being Aryeh Dvoretzky and Menahem Max Schiffer.
Let us end this short biography by recording Rogosinski's debt to Fekete :-
Fekete's genuine love of mathematics showed also in his keen interest in the work of other, and in particular younger, mathematicians. I myself met him first in 1923 at a conference at Innsbruck when his interest in some early work of my own was so encouraging to me at the beginning of my career. My case is not isolated, and in this way he has made himself many life-long friends.
Article by: J J O'Connor and E F Robertson