**Alfréd Haar**'s parents were Emma Fuchs and Ignatz Haar. He was brought up in Budapest which at this time was a flourishing city. 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. Most importantly for the young boy, he was growing up in a city which was a centre for education, and intellectual and artistic life.

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. In 1894 the school teacher Dániel Arany started editing the *Középiskolai Matematikai Lapok*, a mathematical problem solving journal for secondary school students. Each issue of the *Középiskolai Matematikai Lapok* contained a number of selected exercises from mathematics and shortly thereafter from physics, as well as solutions to the past months' problems and a list of those pupils who had sent in correct solutions. Haar collaborated on the journal during his final years at the Gymnasium. In 1903, his final year at the high school, he won first prize in the Eötvös contest in mathematics. This was a defining moment for him for only at this stage did he decide that he would give up his intention of studying chemistry at university and that he would instead pursue a university course in mathematics.

Haar travelled to Germany in 1904 to study at Göttingen and there, after his undergraduate years, he undertook research under Hilbert's supervision. He obtained his doctorate in 1909 with an important dissertation entitled *Zur Theorie der orthogonalen Funktionensysteme* Ⓣ. The main results of his thesis appeared in a paper which he published in *Mathematische Annalen* in the following year. Haar asked a series of fundamental questions about systems of orthonormal functions on the interval [0, 1]. 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 examined the standard systems of orthonormal trigonometric functions and also orthonormal systems related to Sturm-Liouville differential equations. 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. He was appointed as an extraordinary professor at the Franz Josef Royal Hungarian University in Kolozsvár (which is now Cluj in Romania), then in 1917 he became an ordinary professor there being appointed to one of the two chairs of mathematics in the Faculty of Mathematics and Sciences. Of course World War I had broken out two years after Haar took up his extraordinary professorship in Kolozsvár, and times were very difficult in Hungary. Austro-Hungary was aligned to the Central Powers and during the first three years of the war over one million Hungarians died (in addition to two million Austrians) while, for those not fighting on the front, there were food shortages and high inflation. After capitulating on 3 November 1918, Hungary sought a separate peace, independence from Austria, and proclaimed the country a republic. For a short while the country was ruled by a Communist Government but Romanian troops invaded the country and the government was overthrown. In 1921 the Treaty of Trianon was signed which treated Hungary very severely. Its territory was reduced to only one third of its previous size. The terms of the Treaty meant that Kolozsvár was no longer in Hungary (it became part of Romania), so the University there had to move within Hungarian borders. At first it was sited in Budapest for a temporary period of two years before it moved to Szeged, where there had previously been no university. The Department of Mathematics, consisting of the Mathematical Seminary and the Institute of Descriptive Geometry, began operating in Szeged. 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. With support from the Society of Friends of the Franz Josef University, they had founded the famous journal *Acta Scientiarum Mathematicarum* in 1930. 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. They also established the mathematical library again in 1930 and this received mathematics journals from many parts of the world given in exchange for the *Acta Scientiarum Mathematicarum* .

Most of Haar's work was in analysis. After the work of his thesis, which we gave some details of above, he went on to study partial differential equations with applications to elasticity theory. He also wrote on Chebyshev approximations of functions, linear inequalities, analytic functions, and discrete groups. Between 1917 and 1919 he worked on the variational calculus, proving Haar's Lemma, and applying his results to problems like Plateau's problem [1]:-

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. His famous paperA multitude of papers by others show the influence this lemma exerted on the whole area of variational calculus.

*Der Massbegriff in der Theorie der kontinuierlichen Gruppen*Ⓣ appeared in the

*Annals of Mathematics*in 1933. 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. The following celebrates Haar's achievement:-

Haar died in 1933 at the age of 48 years, having been honoured two years earlier by election to the Hungarian Academy of Sciences. In 1975 Mikolás wrote the paper [8] in which he discussed the work of Fejér, Marcel Riesz, Frigyes Riesz, and Haar. He looks at the question:-Said a mathematician named Haar,

"Von Neumann can't see very far.

He missed a great treasure -

They call it Haar measure -

Poor Johnny's just not up to par."

A memorial relief portrays Haar and Riesz in the National Pantheon in Cathedral Square, Szeged. It describes them as:-How did the work of a few great mathematicians contribute to the present high level of analytical investigations, to their applications, and, in general, to the intensive mathematical life of today in Hungary?

... the world-famous founders of the Szeged mathematical school.

**Article by:** *J J O'Connor* and *E F Robertson*

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