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Marcel Riesz's doctoral thesis

Marcel Riesz's thesis advisor at the University of Budapest was Lipót Féjér. He wrote his doctoral thesis Összegezhető trigonometrikus sorok és összegezhető hatvány-sorok (1908) in Hungarian having published a short report on the main results in French in Comptes Rendus of the Academy of Sciences with the paper Sur les séries trigonométriques (7 October 1907). He published a substantially revised and expanded version of his thesis in German in Über summierbare trigonometrische Reihen (1911).

We give below an English translation of the introduction to Marcel Riesz's 1908 doctoral thesis and also an English translation of the introduction to Über summierbare trigonometrische Reihen (1911).

1. Summable trigonometric series and summable power series (1908).

Since the investigations of Du Bois-Reymond we know that the points of divergence of the Fourier series of a continuous function may form an every-where dense set. Thus not even continuous functions have the property that their Fourier series converges.

The recent tendency in the theory of functions is not exhausted by merely looking for convergence criteria but searches for other connections between the function and its Fourier series. Fejér was the first to apply consciously this far-reaching point of view in the theory of these series. He recognised that the method of summation, applied by Cesàro and Frobenius to other questions (which, instead of the limit of the partial sums, investigates the limit of their arithmetic means), is of capital importance for the Fourier series.

The divergent series which are summable by means of this method belong at any rate to the divergent series which are the easiest to handle and therefore the most useful. A series of this kind will be said to be summable in brief, and the limit of the arithmetic means will be called the Cesàro-sum of the series.

Fejér's main result is the following:

The Fourier series of any finite and integrable function f(x)f(x) is summable at every point where the function is continuous or has a discontinuity of the first kind, the value of the Cesàro-sum is f(x)f(x) at the points of continuity and 12[f(x+0)+f(x−0)]\large\frac{1}{2}\normalsize [f(x+0)+f(x-0)] at the points of discontinuity of the first kind.

This penetrating theorem was significantly generalised later from different points of view by Fejér himself and by Lebesgue.

It is also an important property of Fejér's summation that the arithmetic means converge uniformly to the values of the function in any interval of continuity of the function, while, as it was recently shown by Lebesgue, the Fourier series of an everywhere continuous function may converge everywhere without being uniformly convergent.

It is known that the Fourier series are special trigonometric series, whose coefficients are given by the familiar Fourier integral formulas. Thus when we consider these series we always start from a given function. The most important theorems in the theory of convergent trigonometric series given a priori were given by Riemann, Cantor and Du Bois-Reymond. Their results will occur during our discussions. Now I single out only one of the most typical among them, Cantor's theorem. This reads as follows:

If a trigonometric series converges in the interval (0, 2π) (except possibly for a reducible set of points) and its sum vanishes, then also all its coefficients vanish.

Cantor's proof is based on theorems of Riemann and Schwarz.

We saw above on the example of Fourier series that the method of summation by arithmetic means suits the nature of trigonometric series better than forming the ordinary sum. Thus the question arises naturally whether Cantor's theorem can be generalised to divergent but summable trigonometric series. In his doctoral dissertation, Fejér is still sceptical about this question, since the Cesàro-sum of the divergent series
12+cos⁡x+cos⁡2x+...+cos⁡nx+...\large\frac{1}{2}\normalsize + \cos x + \cos 2x + ... + \cos nx + ...
is zero at every point of the interval (0,2π), except at the points 0 and 2π, nevertheless not all the coefficients of this series do vanish. In the "Mathematische Annalen", however, Fejér already extends Riemann's theorem used by Cantor thus he so-to-say prepares the answer to be given to this question.

During our common stay in Göttingen, Mr Fejér himself was kind enough to call my attention to this subject, for which it is my pleasant duty here to thank him.

This work is divided into two parts. In the first part I establish among others the following theorems:
If the series
∑n=1∞(∣an∣+∣bn∣)/b2\sum _{n=1}^{∞}(|a_{n}| +|b_{n}| )/b^{2}
formed with the coefficients of some trigonometric series is convergent, then the Cesàro-sum of this trigonometric series can vanish at every point of the interval (0,2) only if all its coefficients vanish.

If the coefficients of a trigonometric series tend to zero, and if in the interval (0,2π), with the exception of a reducible set, the series is summable and its Cesàro-sum is zero, then all its coefficients vanish.

For the proof of our theorem we had to generalise Schwarz's mentioned theorem in a suitable way. Our result reads as follows:
Every continuous function, whose generalised second derivative is continuous, and whose generalised fourth derivative is zero, is a polynomial of degree three.
This result, together with Fejér's generalisation of Riemann's theorem, and furthermore the solution of a functional equation, lead immediately to our first theorem.

To prove our statement concerning series with coefficients tending to zero, we had to deduce an additional simple lemma.

As a consequence of these investigations, we present then the analogue of Du Bois-Reymond's fundamental theorem, which we prove with the help of a mean-value theorem, interesting in itself.

In the second part of our article, we investigate the behaviour of some power series on their circle of convergence, completely independently of the first part.

Our investigations enable us to form strongly divergent summable series, and on the other hand they indicate the applicability of the iterated summation process. Namely, we present certain classes of power series which are summable by the nn times iterated summation process, while a number less than n does not achieve the goal. As far as we know, this has not yet been done.

Further on we extend our results to the case when the order of summation is not an integer. At this point our investigations are related to some results of Hadamard.

2. On summable trigonometric series (1911).

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?

As is well known, the uniqueness problem of convergent trigonometric series was solved by Cantor's theorem, which is essentially based on two theorems by Riemann and one by Schwarz.

The question of the analogue of Cantor's theorem first arises in Fejér's dissertation. He is sceptical of the question here, since the well-known series
12+cos⁡x+...+cos⁡nx+...\large\frac{1}{2}\normalsize + \cos x + ... + \cos nx + ...
can be summed at every point x≢0(mod  2π)x \not\equiv 0(\mod 2 \pi) using his method and has the value zero. However, in his Annalen paper, he already gives a far-reaching analogue of Riemann's theorem, thus paving the way for the solution of the problem.

A conversation with Mr Fejér on this matter in the summer of 1907 prompted me to take up the problem. I have proven that, in general, the development is indeed unique. My proof, besides the aforementioned Fejér theorem, is based on a new mean value theorem, from which an analogue of Schwarz's theorem then follows. In order to apply these theorems, I had to subject the trigonometric series under investigation to a certain restriction.

The special position of Fejér's example above is explained by the fact that, in our case, unlike the original Cantor theorem, exceptions cannot generally be permitted.

Similar to Cantor's theorem, Du Bois-Reymond's theorem can also be extended to summable trigonometric series.

At the end of our work, we will discuss arithmetic means of arbitrary order - integer or non-integer. The order 1 will, in a certain sense, present itself as a limiting case. While the two classical theorems mentioned above can be extended if the order of the summation is kk ≤ 1 (for k = 1 with the indicated restriction), this is certainly not the case for kk > 1.

The resulting similarities and contrasts also offer some new perspectives on the original theorems.
Last Updated March 2026