It is well known from the work of Fejér what an outstanding role the means of the partial sums play in various problems concerning Fourier series.

Riemann juxtaposed the theory of Fourier series with the theory of trigonometric series formally given by their coefficients. The convergence theory of these series is linked to the names of Riemann, Cantor, and Du Bois-Reymond.

Fejér's results on Fourier series suggest investigating to what extent the theorems of the aforementioned researchers retain their validity if, in the theory of formally given trigonometric series, only summability by means of the partial sums is assumed instead of convergence.

The first question is one of uniqueness: Can a function be represented by two different summable trigonometric series?

Lund did not have much of a mathematical tradition but Riesz's arrival meant a change of atmosphere. He was now an international star, active with his own research and he also had the time and incentive to broaden his interests.

The publication of this tract has been delayed by a variety of causes, and I am now compelled to issue it without Dr Riesz's help in the final correction of the proofs. This has at any rate one advantage, that it gives me the opportunity of saying how conscious I am that whatever value it possesses is due mainly to his contributions to it, and in particular to the fact that it contains the first systematic account of his beautiful theory of the summation of series by 'typical means'.

The task of condensing any account of so extensive a theory into the compass of one of these tracts has proved an exceedingly difficult one. Many important theorems are stated without proof, and many details are left to the reader. I believe, however, that our account is full enough to serve as a guide to other mathematicians researching in this and allied subjects. Such readers will be familiar with Landau's 'Handbuch der Lehre von der Verteilung der Primzahlen' Ⓣ, and will hardly need to be told how much we, in common with all other investigators in this field, owe to the writings and to the personal encouragement of its author.

To the consummate skill in handling formulas which is typical of the classical Hungarian school he now added a more abstract, functional analytic view, long before functional analysis had become a commonplace tool.

... contains an account of all results previously published without detailed proofs. Formally, therefore, we have a research paper. However, the paper is written in the style of a book. All details are given and, where desirable, passages of previously published papers are reproduced. The author takes care to compare his method with others and to point out the details which make the mechanism work. The paper is self-contained and should be accessible also to non-experts.

In 1949, Professor Riesz published a long paper in which he introduced a multiple integral of Riemann-Liouville type and showed how important this idea is in the theory of the wave equation. By this means he gave the solution of the problem of Cauchy for a space-like boundary and also the solution of a characteristic boundary value problem when the unknown function is given on a characteristic half-cone.

He now gives the solution of the wave equation for a very general class of characteristic boundaries. In the three-dimensional case, a specialisation of the solution leads to an interesting case of curvatura integra for twisted curves. The differential-geometrical consequences of the formula obtained are examined independently.

It is known that discontinuities of a solution of a partial differential equation can occur only on characteristic surfaces of the equation. In the last section a sequence of special functions is constructed which are related to a given characteristic surface in the same way as the powers of the Lorentz distance are related to a characteristic cone. In particular, the author gives an explicit solution of the wave equation which is infinite on the characteristic surface in the same way as the elementary solution is on a characteristic cone.

The whole presentation is very elegant.

M Riesz was a perfectionist and a gentleman. He spent endless trouble on giving his presentation a high polish and he worked as hard on improving the proofs of known theorems as on the finding of new ones. The final publication of his results often took place many years after their first announcements. His masterful exposition makes his papers a joy to read. His work covered a wide area of analysis and just an enumeration would be too long for a review.

He then had a small circle of graduate students. Each one got personal attention. Riesz loved to talk about mathematics and he appreciated having listeners. He could go on for hours and when he was in good form, his grip on the listener never slackened. Riesz lived alone and these personal lectures took place sometimes in his home, sometimes in his favourite café and sometimes over the telephone. ... He worked constantly, often at late hours and periodically with great intensity. These habits did not change much with advancing age ...

His pedagogical successes can be explained above all by the fact that he was always ready to talk about mathematics with anyone, at any time. He liked to get up late, and then work tirelessly, with fantastic energy, until dawn the next day, or even until morning. In June 1949, we travelled from Paris to Grenoble at night, on a crowded train, there was no question of sleeping, of course. Marcel Riesz, standing by the window, was looking out; Alfred Rényi asked him what he was thinking about. "Dirichlet's problem," he replied. That same summer, Jacques Deny came from Strasbourg to Paris to meet him. Deny was writing his doctoral thesis on potential theory, a thesis which, to a large extent, extended Frostman's work, placing it on new foundations. Riesz held Deny's work in high esteem and declared that it played the same role in relation to Schwartz's theory of distributions as Fatou's thesis, dating from 1906, did in relation to Lebesgue's then-new notion of the integral. So Deny arrived in Paris, met Riesz at noon, and until dawn the next day they talked without interruption about potential theory. That same day, poor Deny, of fragile constitution, was already returning to Strasbourg and declaring that he would not survive another encounter with Marcel Riesz.

Indiana University becomes one of the world's leading centres for research and teaching of relativity with the addition to its mathematics staff of Marcel Riesz, Hungarian-born Swedish mathematician. He joins another famous expert on relativity, Vaclav Hlavaty, who has been a member of the I.U. faculty since 1948: Riesz will remain for five months, substituting for Prof Eberhard Hopf, who is on leave of absence. Both Riesz and Hlavaty turned their attention to relativity after attaining eminence in the same field, differential geometry. They are applying geometrical principles to the solution of specialised problems in relativity.

The seminal material which contributed greatly to the start of modern research on Clifford algebras is supplemented by notes which Riesz dictated to Folke Bolinder in the following year and which were intended to be a fifth chapter of the Riesz lecture notes.

As a young student, he wrote and spoke German and French remarkably well, but he only began to study English just before his visit to Chicago, that is, at a relatively late age. Despite this, he very quickly achieved a perfect and refined command of the language. While until 1955, apart from 'The general theory of Dirichlet's series' with G H Hardy, he published nothing in English (and even then, it was Hardy who wrote the text for 'The general theory of Dirichlet's series'), from 1955 onward he published only in English.

... bore the burden of his last illness with great courage.

Apart from mathematics, Marcel Riesz had many interests; in particular, he followed current events with great attention. In his youth, he loved the theatre and, thanks to his prodigious memory, he was still able, many years later, to recite, or even act out, entire sequences from the plays he had seen. It was undoubtedly also thanks to his astonishing memory that, although he spoke five languages ​​perfectly, he continued until the end of his life to speak impeccable and flavourful Hungarian, without the slightest trace of that fault, all too common among Hungarian émigrés, of peppering their speech with foreign words. He only had to hear a proof once; he never forgot it, and thanks to this, a fine estimate by Leopold Fejér was saved from oblivion.