**Alfréd Haar** was a Hungarian mathematician who is best remembered for his work on analysis on groups, introducing a measure on groups, now called the *Haar measure*.

- After the unification of Buda, Obuda and Pest in 1872, thirteen years before Haar's birth, the city became not only the capital of Hungary but also a major centre for industry, trade, communications, and architecture.
- Haar attended the Gymnasium in Budapest and, as might be expected, he was an outstanding student showing great potential for science.
- For most of his time at the Gymnasium Haar felt that chemistry was the subject for him but he also did outstanding work in mathematics.
- Haar collaborated on the journal during his final years at the Gymnasium.
- Haar travelled to Germany in 1904 to study at Göttingen and there, after his undergraduate years, he undertook research under Hilbert's supervision.
- Haar wrote: one wants to be able to determine sufficient conditions that a series of such functions is convergent; one wants examples of relatively sensible functions which do not converge in the pointwise or uniform sense; one wants to understand how summation methods may be used to overcome the problems of divergence; and one wants to know exactly when, if the series of partial sums of an orthogonal expansion of a function converges, its limit equals the original function.
- He constructed what is now known as Haar's orthonormal basis to answer the question of divergence of continuous functions expanded as series of orthonormal systems of functions.
- The paper introduced to the mathematical world what are today called Haar wavelets, an orthogonal system of discontinuous functions admitting at most three values, which Haar had first introduced in an appendix to his doctoral thesis.
- Haar was appointed as a privatdozent at the University of Göttingen immediately after completing his doctoral thesis, and he taught there until 1912 when he returned to Hungary.
- The department consisted of Riesz, Haar, Rudolf Ortvay (who held the Chair of Mathematical Physics), and Tibor Radó who had been appointed as Haar's assistant.
- There were no other assistants in the department at this time but István Lipka became Haar's assistant in 1926.
- Haar, together with Riesz, rapidly made a major mathematical centre from the new university.
- Haar and Riesz were the editors and the reputation of the journal was quickly established with mathematicians of the quality of John von Neumann, Norbert Wiener, George D Birkhoff, Henri Cartan, Antoni Zygmund, George Pólya, Paul Erdős (still a student at the time) publishing a paper in the first volume.
- Most of Haar's work was in analysis.
- Haar is best remembered, however, for his work on analysis on groups.
- In 1932 he introduced an invariant measure on locally compact groups, now called the Haar measure, which allows an analogue of Lebesgue integrals to be defined on locally compact topological groups.
- The concept of Haar was used by von Neumann, by Pontryagin in 1934, and Weil in 1940, to set up an abstract theory of commutative harmonic analysis.
- At first, however, von Neumann tried to discourage Haar in seeking such a measure since he felt certain that no such measure could exist.
- A memorial relief portrays Haar and Riesz in the National Pantheon in Cathedral Square, Szeged.

Born 11 October 1885, Budapest, Hungary. Died 16 March 1933, Szeged, Hungary.

View full biography at MacTutor

Origin Hungary

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive