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In 1897 Fejér won a prize in one of the first mathematics competitions to be held in Hungary. From that year until 1902 Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin where he was a student of Schwarz . After he changed his name from Weiss to Fejér, Schwarz refused to talk to him!

In 1900 Fejér published a fundamental summation theorem for Fourier series . This work was the basis of his doctoral thesis which he presented to the University of Budapest in 1902.

From 1902 to 1905 Fejér taught at the University of Budapest and from 1905 until 1911 he taught at Kolozsvár in Hungary (now Cluj in Romania). In 1911 Fejér was appointed to the chair of mathematics at the University of Budapest and he held that post until his death. However there were problems regarding his appointment to the chair as is recounted in :

Although already world famous and warmly endorsed by Poincaré on the occasion of the awarding of the Bolyai Prize, Fejér's appointment to a chair at the University had been opposed by anti-semites on the Faculty. One of them, knowing full well that Fejér's original name had been Weiss, asked during the occasion of Fejér's candidacy: 'Is this Leopold Fejér related to our distinguished colleague on the Faculty of Theology, Father Ignatius Fejér?' Without missing a beat Loránd Eötvös , Professor of Physics, answered "Illegitimate son". After that the appointment sailed through smoothly.

During his period in the chair at Budapest Fejér led a highly successful Hungarian school of analysis.

Fejér's main work was in harmonic analysis . He worked on power series and on potential theory. Much of his work is on Fourier series and their singularities but he also contributed to approximation theory.

Fejér collaborated to produce important papers, one with Carathéodory on entire functions in 1907 and another major work with Riesz in 1922 on conformal mappings .

One of Fejér's students described his lecturing style in the following way (see ):

Fejér gave very short, very beautiful lectures. They lasted less than an hour. You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression. The lecture was thought out in very great detail, with dramatic denouement. He seemed to relive the birth of the theorem; we were present at the creation. He made his famous contemporaries equally vivid; they rose from the pages of the textbooks. That made mathematics appear as a social as well as an intellectual activity.

Source:School of Mathematics and Statistics University of St Andrews, Scotland