MARCEL RIESZ IN MEMORIAM

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MARCEL RIESZ IN MEMORIAM

B Y

LARS G A R D I N G

L u n d

Marcel Riesz was born in GySr, H u n g a r y , N o v e m b e r 16, 1886 and died in Lund, Sweden, on September 4, 1969. H e studied in Budapest, GSttingen a n d Paris. I n 1911 Mittag-Leffler invited him to come to Sweden where he t a u g h t at Stockholms HSgskola.

I n 1926 he was appointed professor of m a t h e m a t i c s at the university of Lund. After retiring from this position in 1952 he went to the United States where he was visiting research professor at the universities of MaryIand a n d Chicago and other places. H e returned to L u n d in 1962. He was a m e m b e r of the Swedish A c a d e m y of Sciences, the Physiographical Society in Lund, Videnskabernes Selskab in Copenhagen, had honorary degrees from the universities of Copenhagen and Lund a n d was an honorary m e m b e r of the Swedish mathematical society.

Marcel Riesz was the youngest m e m b e r of a generation of brilliant H u n g a r i a n mathematicians t h a t included among others Leopold Fejdr, Riesz's elder brother Frederick Riesz and Alfred H a a r . His first p a p e r ([1], 1906) is an exposition in H u n g a r i a n of a subject of current interest at the time, namely various s u m m a t i o n methods for Taylor series of analytic functions. One of these methods, due to Mittag-Leffler, sums the series in a star- shaped region bounded b y singular points, the Mittag-Leffler star. Common interest in

0 -- 7 0 2 9 0 9 . A c t a mathematioa. 124. I m p r i m $ le 7 A v r i l 1970.

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ii LARS OAXDI2r

these matters was the beginning of the association with Mittag-Leffler. They seem to have met for the first time in Stockholm in 1908.

I n the period from 1908 to 1916, Riesz did very successful work in the summability theory of power series, trigonometric series and Dirichlet series. The starting point was Fejdr's discovery t h a t the Fourier series of a function / is summable by arithmetic means to/(x) at every point of continuity. Lcbesgue had shown t h a t the same holds at all Lebesgue points. The current scales of summation methods were those of HOlder and Ces~ro, recently shown to be equivalent. The Ceshro mean s(n ~) of order ~ > - 1 of a sequence {s~)~ ~ s o = 0, or a series with partial sums s~ is by definition the quotient of the coefficients of the n t h terms in the formal power series for ( 1 - x) -~ ~ o s~x n and ( 1 - x) -~-I. The Ceshro limit is lira s(= ~) for n-~ oo. When ~ = 1, this procedure sums the series by arithmetic means and the result of Lebesgue applies. One of the classical results in Fourier series, namely t h a t Fej4r's (and Lebesgue's) theorem holds for summatioll of order ~ > 0 is essentially due to Riesz. He announced it in 1909 [7] and published his proof much later [25]. I n the meantime there were proofs by Chapman in 1910-11 and by H a r d y in 1913. Riesz's main contribution, however, was in another direction. He observed that Ces~ro's summation is equivalent to another one which has the advantage of being applicable also to functions s(x) of locally bounded variation for x ~> 0 such t h a t s(0)=s(0 + ) = 0. I t s means are

sa(x) = x -~ (x - t ) a d s ( t ) (1)

and the limit is defined accordingly. I n case of a series we put s ( x ) = s ~ when n < ~ x < n + l . The right-hand side of (1) appears also in the classical l~iemann-Liouville integral of a f u n c t i o n / ,

I ~ / ( x ) = ~ (x - t)~-l/(t) dr. (2)

The semi-group properties of the operator a-+ I~, namely that I ~ IZ = I ~+~ and I ~ = identity has immediate application to the means (1). The corresponding summation method applies to generalized Dirichlet series

f(z) = f / e - ~ ds(t),

where z is a complex number. The means are

f

x -~ e - ~ t ( x - t F d s ( t )

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MARCEL R I E S Z I N MEMORIAM

and the a-sum of

/(z)

is obtained b y letting x-+ ~ . These integrals were introduced by Riesz in 1909 [7] under the name of typical means. They had an instant success and led to a Cambridge Tract on Dirichlet series 1915 written jointly by H a r d y and Riesz and reprinted in 1952.

Some ten years later, Riesz had established himself as an expert in the theory of trigonometric series and together with E. Hilb wrote the article Neuere Untersuchungen fiber trigonometrische Reihen in Eneyklop/~die der Mathematischen Wissenschaften. I t would carry us too far to penetrate in detail Riesz's contributions to this intricate and much studied subject, b u t at least three more things s t a n d o u t . His interpolation formula [14] for trigonometric polynomials has a permanent place in interpolation theory and provides simple proofs of, e.g., the well-known inequalities of Bernstein and Markoff with the best constants.

One recurrent theme in Riesz's work is a theorem by F a t o u that says t h a t the powerseries of an analytic function converges at every point of the boundary of the circle of convergence where the function is regular and the coefficients of the series tend to zero.

Riesz gave a beautiful proof [17] of this result that also showed t h a t the convergence is locally uniform on every are of regularity. This proof is available in Landau's little booklet

Darstellung und Begriindung einiger neuerer Ergebnisse der Funktionentheorie.

Riesz also extended F a t o u ' s theorem to Dirichlet series and to summability situations. His best known result in Fourier series, however, lies in another direction. E v e r y trigonometric series with real coefficients can be written as

u(e ~~ = ~

Re

c~e ~n~ .

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0

Formally, this is the real part of the power series I(~) = >1 c,,~", z =,-~'

0

for r = 1. Its imaginary part

v(e i~ = ~.

I m c~ e ~~

0

is called the conjugate series to (3). By Parseval's formula

f ;~ u( e'O)2 dO = ~r ~ , cn ,2 = f ~ v( e'~ dO ,

(4) provided we normalize so t h a t c o =0. Hence

u--->v

is an isometry

L ~ L ~

where L 2 refers to the interval 0,2•. The connection between conjugate series was an early subject of study.

I n particular, as early as 1912, W. H. and G. C. Young had considered the question whether the m a p

u-+v

might be, if not an isometry, then at least a homeomorphism L V ~ L ~,

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I V L A R S G ~ R D U R G

1 < p < co, but t h e y had some indirect evidence for a negative answer. Here was an outstand- ing unsolved problem and when Riesz wrote to H a r d y in 1923 t h a t he had found a proof (see [33]), H a r d y wrote back "some months ago you said 'j'ai d~montrd que 2 s~r. trig.

conjugu4es sont toujours en m~me temps les s~ries de F de fonctions de la classeLP(p > 1)...'.

I want the proo[. Both I and m y pupil Titchmarsh have tried in vain to prove i t . . . " . And in the next letter "Very m a n y t h a n k s - - y o u supply all t h a t is essential, I have sent on your letter to Titchmarsh. Most elegant and beautiful. Of course p. 2 is the real point. I t is amazing t h a t none of us should have seen it before (even for p = 4!)". W h a t H a r d y is refering to is Riesz's use of Cauehy's formula which gives ~ " ](e~a)~dO =0 when p is an integer.

Taking the real part and putting p = 2 proves (4) while, if p = 4,

~ (u 4 ^- 6 u~v ~ + v 4) dO = O.

This identity and a moment's reflection show t h a t ~0 ~" vPdO <~ c~,~ "~ u~dO when p = 4 and, more generally, when p is an even integer. The remaining arguments of the proof were more technical but t h e y eventually lead Riesz to a basic result in analysis, his convexity theorem [32]. I t combines elementary form with great power and runs as follows: the logarithm of the maximum 2~I~ of the absolute value of a bilinear form

under the conditions

J = l k = l

j ~ l k = l

is a convex function of ~, fl in the triangle

A little later, Riesz's student Olof Thorin proved convexity when a ~>0, fl >~0 provided the variables x and y are allowed to be complex. The convexity has m a n y applications, one of them to Riesz's theorem about conjugate functions. Let cp be the best constant in the inequality [[vll s ~< %[lull ~ where I1[[1 ~ = (S~l/(e'~)[ ~dO)I/p" Then, b y H61der's inequality, % is the supremum of the absolute value of the bilinear form A(u, w) = . ~ u(e~O)w(etO)dO when Iluil~<l and ilwllo~<l where p-1 + q - l = l . Further, c~=%. Hence, since c2=1 and, by the argument given above, c~ < oo when p is an even integer, it follows t h a t % < oo when 1 < p < ~ . Riesz's convexity theorem has been the model for similar results, in recent times also for the theory of interpolation spaces.

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M A R C E L R I E S Z I N MEMORIAM

Before Riesz m o v e d to Lund he also worked with the m o m e n t problem ([24]), in particular the H a m b u r g e r m o m e n t problem: find a real measure d#>~0 on the real line with given m o m e n t s

c~ = x~ d~u( x).

H a n s H a m b u r g e r ' s almost complete analysis of this problem in 1920 a t t r a c t e d the a t t e n t i o n of m a n y mathematicians, among t h e m Riesz and Torsten Carleman. H e r e Carleman, who could fit the problem into his theory of singular integral equations, was perhaps more successful. Riesz was prepared for the subject through his friend Erik Stridsberg who took a lively interest in Stieltjes's classical paper from 1894 a b o u t the m o m e n t problem on the ha/f-axis x >/0.

The m e a t of the m o m e n t problem is the analysis of the set of solutions and the problem of uniqueness. Existence of a solution # is a relatively simple matter. I t is obviously necessary that, for every real polynomial P,

P(z) =~a,~x'>~O ~ ~a,~cn>~O*

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and it is easy to see t h a t this is equivalent to the condition t h a t all the quadratic forms

~'~ ~ Cj+kyjyk

are ~> 0 . To prove t h a t (5) is also sufficient one extends the non-negative linear functional ~0(P)=F~anc,~ from polynomials to the space of real continuous functions of at m o s t polynomial growth. This procedure is now familiar to every student of analysis who knows the H a h n - B a n a c h theorem, but it is interesting to note t h a t Riesz used it in 1918 (see [24], the third note p. 2) in a lecture to the Stockholm Mathematical Society, well before Banach (1923, 1929) and H a h n (1926).

Riesz's work after he m o v e d to Lund m a r k s a b r e a k with the past. H e acquired new interests, starting work in potential t h e o r y a n d wave propagation including Dirac's equation of the electron a n d relativity theory. H e also took a continuing interest in e l e m e n t a r y n u m b e r theory. His most i m p o r t a n t contributions are in potential theory and wave propagation. I n b o t h cases he invented new multi-dimensional analogues of the Riemann- Liouville integral (2). The one used in potential theory is the fractional potential of order of a mass d i s t r i b u t i o n / d x ,

I~l(x) = Hm ((z)-l f l

^{x -}

y

I ~ ' - ~

t(Y) dy. (6)

H e r e I x - y ] is the euclidean distance in R m, 0 < a < m, and

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vi LARS a~DrNG

iS chosen so t h a t I ~ ] ~ = ]~+~ when a > O, ~ >0, a + ~ < m . B y a passage to the limit, I ~ is the identity. I n particular, if

U~ f ( x ) = Hm ( a ) - l f l x - y [~-'n dff(x)

is the a-potential of a mass distribution ff one has

whcrt a > 0 , ~ > 0 , 0 r ~ < m . Hence

f u~ ff(x) df(x) = HAa) f ( u~'ff(~))* dx*

⁽⁶⁾

is non-negative. When a = 2 < m , this represents the self-energy of the mass distribution # with respect to the Newtonian potential. I n this case the equilibrium distribution ffF on a compact set F had been characterized by Gauss as having minimal energy in the class of mass distributions # supported by F and having a given total mass. After his student Otto Frostman had put this principle on a strict basis, Riesz noted t h a t the formula (6) indicates t h a t the equilibrium distribution with respect to an a-potential could be defined b y this principle of Gauss. I n the case 0<a--<2, which is the condition for the kernel I x - y l ~ - " to be subharmonic, this program met with complete success in Frostman's well-known thesis. The equilibrium distribution ffF exists, it is unique, its potential on F is essentially constant, there is a notion of a-capacity and there are generalizations of the balayage process and Green's function. After Frostman, Riesz himself wrote on the subject [42]. Viewed in perspective, these contributions opened the way to the present-day variety of generalizations and refinements of classical potential theory.

At some points in his work on a-potentials, Riesz uses the fact t h a t if ] is smooth enough, (6) is a regular analytic function of a when c < R e a < m with a given c and t h a t A1 ~= I ~-2 where A = (~/~xl)2+ .... + (~/~xm) ~ is Laplace's operator. Analyticity properties became very important in the integral

= H m ( a ) - l _ y ) ~ - ,~ ] ( y ) d y

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associated w i t h the wave operator

A = ( ~ l ~ x ~ ) ~ - . . . - ( ~ / ~ x , ~ ) ~.

Here r ~ ( x ) = x ~ - ... - x ~ is the square of the Lorentz distance and C:r~(x)>70, x I > 0 is the

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M A R C E L R I E S Z I N M E M O R I A M v i i

forward light-cone so t h a t x - C is the retrograde light-cone with its vertex at x. Finally,

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Clearly, (7) is an analytic function of ~ when the integral converges, i.e. when Re ~ > m - 2 . The factor H a is chosen so t h a t

A I ~ = I ~-2, I ~ I ~ = I ~+~ (9)

for large ~ and ft. The main point is now t h a t , if / is smooth enough, the right side of (7) is analytic in the halfplane Re zr > - c with a n y given c, t h a t (9) holds under corresponding circumstances and t h a t I ~ = 1. The real explanation for this is perhaps the following. Let i t have compact support and let

d x

be the Fourier-Laplace transform of it. Then

i it(x) = (2

u)-m f e=("'" ) + +

where the integral is independent of the choice of ~ ~ C and the right side is indeed entire analytic in ~ when it is infinitely differentiable. Although Riesz was conscious of this, he always preferred direct proofs avoiding the use of the Fourier-Laplace transform. I n his last published p a p e r [59] he gives a v a r i a n t of an earlier direct proof t h a t I ~ is the identity.

This point of view is connected with his intense interest in special relativity and questions of invariance under the Lorentz group.

I t is probable t h a t Riesz conceived the integral (7) during a series of seminars in the beginning of the thirties conducted b y his colleague Nils Zeilon. The subject was H a d a - m a r d ' s theory of finite parts of integrals as exposed in his well-known book on the Cauchy problem. H a d a m a r d gives explicit formulas for the solution of Cauchy's problem with d a t a on a space-like surface for the wave equation with variable coefficients. I n 1938 Riesz was r e a d y with a beautiful v a r i a n t of these formulas based on an operation I ~ analogous to (7) a n d with similar properties relative to the wave operator with variable coefficients.

H e published a n account in a note added to a lecture [43] given a ),ear earlier where he treats the case of constant coefficients. I n the lecture he remarked among other things t h a t the fact t h a t 1/Hm(2)=0 when m > 2 is even reflects H u y g e n s ' principle and he finished b y giving an invariant formula for the solution of Cauehy's problem when m = 4 and analyzing it geometrically. I n the fifties, during his s t a y in the United States, Riesz devoted m u c h

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VIII L ~ S G ~ D ~ G

t h o u g h t to this formula and e x t e n d e d it to characteristic surfaces in a n intriguing paper [55].

The short p a p e r [43] contains the main body of Riesz's work on the wave equation a n d it still makes impressive reading. The detailed account came in 1949 in a massive p a p e r [47] in the Acta Mathematica. Considering t h a t Riesz was a n a t u r a l perfectionist and, in his later years, a slow worker, this p a p e r is the result of an enormous effort on his part. I t was made possible only through collaboration with his friend K. G. H a g s t r o e m who graciously offered his hospitality and his services as a secretary during m a n y summers in the forties.

Riesz's work on the wave equation did not contribute to the existence t h e o r y of Cauchy's problem and one of its main goals, the proper interpretation of divergent integrals, has since been attained in a m u c h larger f r a m e w o r k through L a u r e n t Schwartz's theory of distributions. B u t the m e t h o d of analytical continuation which makes sense also in this theory, is a lasting contribution.

Around 1930, the teaching of q u a n t u m mechanics and relativity theory became a necessity a t every self-respecting university, b u t at the time n o t m a n y physicists had the m a t h e m a t i c a l training necessary for these subjects. For m a n y years Riesz lectured periodically on tensors and matrices to a motley crew of physicists and mathematicians. He also s t a r t e d research of his own and published his first p a p e r in relativistic q u a n t u m theory in 1946. His m a i n interest was the Dirac equation and the Clifford algebra. I n [53] he introduced w h a t a m o u n t s to a Clifford structure on a Lorentz manifold. Riesz also applied his work on the wave equation to t h e classical relativistic theory of the electron [49, 50].

The p o p u l a r lecture [44] on the models of non-euclidean geometry, a masterpiece in its kind, is a b y p r o d u c t of his interest in the geometry of relativity.

Riesz ~ r o t e clearly and well and paid m u c h attention to form. His favourite language was French and his style, steeped in the classical tradition, sometimes borders on the pre- cious. Mathematical research always involves competition for fame and a place in the hierarchy, b u t he m a d e it seem a gentleman's game. H e was of course no stranger to ambi- tiort a n d h a d to assert himself b o t h in Sweden and in the cosmopolitan world he came from. H e admired his illustrious brother Frederick and t h e y h a d cordial relations. T h e y wrote one p a p e r [22] together, b u t otherwise there is a clear distance in content between their work, perhaps a result of a conscious effort on Marcel's part. Seen together t h e y had m u c h physical resemblance b u t v e r y different t e m p e r a m e n t s , Frederick calm with great poise and Marcel quick and restless in comparison. Marcel Riesz knew an astonishing number of mathematicians and over the years made a n d kept m a n y friends among them.

Mittag-Leffler h a d m a d e Stockholms tt6gskola a center of m a t h e m a t i c a l research.

I t h a d a p e a k of activity before a n d around the t u r n of the century a n d its other mathe-

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M A R C E L R I E S Z I N M E M O R I A M IX maticians, Bendixson, yon Koch and Fredholm were famous names. Some ten years later, however, their scientific activity was on the wane for ~arious reasons. Riesz filled a mathematical vacuum. I I e learnt Swedish quickly and he was very active in the local mathematical society where he soon became the dominating figure. He was lively, accessible, an enthusiastic teacher and a good lecturer with a thorough knowledge of his field. His charm- ing expository lecture [15] from 1913, written in Swedish, has a distinct personal touch reflecting these qualities. I n 1923, Riesz lost a competition for a chair in L u n d to Carleman.

Shortly afterwards yon Koch died and Bendixson, Fredholm and Phragmdn made a move to appoint Riesz as y o n Koch's successor. The move failed and the call went to Carleman.

Shortly afterwards, Riesz got a position in Lund. At least in the beginning, he must have felt his stay in L u n d as an exile. He had been very successful in Stockholm where, among others, ttarald Cramdr and Einar Hille had been his personal students.

L u n d 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. F r o s t m a n ' s thesis was a success and there were others after him. Lars H6rmander was one of Riesz's last personal students in Sweden. Riesz's work on fractional potentials was the origin of the contributions from L u n d to the theory of partial differential operators.

I met Marcel Riesz in 1937, m y first year at the university. He then had a small circle of graduate students. Each one got personal attention. Riesz loved to talk about mathe- matics 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 car@ and sometimes over the telphone. The subjects were mostly those t h a t occupied him at the time, fractional potentials, relativity theory and the Clifford algebra. He was not inclined to give away thesis subjects, but if somebody had a promising idea he was a very stimulating partner and spared no effort. He worked constantly, often at late hours and periodically with great intensity. These habits did not change much with advancing age and eventually took their toll. After about ten years in the United States he had a breakdown t h a t forced him to return to Sweden and begin a more sedate life.

Although Riesz's interest in administrative matters was strictly limited to those concerning his own subject, he had great influence in the faculty merely by his good sense, his wit and his charm. He was a shrewd judge of personalities and h u m a n affairs and had an u n c a n n y knack for the right word at the right moment. His memory was extraordinary and he was a fascinating teller of stories. Unfortunately he never wrote his memoirs. He had a long and active life and bore the burden of his last illness with great courage.

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x LAaS G~LRD~O

Bibliography

The works of Marcel Riesz in chronological order.

[t]. Representation of the analytical c o n t i n u a t i o n of a given powerseries I, I I (Hungarian).

Math. ds Phys. Lapok, X V I (1906), 1-25, 96-108.

[2]. Summierbare trigonometrische Reihen und Potenzreihen (Hungarian). Thesis, Budapest, 1908.

[3]. The sunmaability of a power series on the b o u n d a r y of the circle of convergence (Hungar- ian). Hung. Acad. Sc. Math. ds Term~szettud. drtisit6 (1908).

[4]. Sur les s~ries trigonometriques. C. R. Paris, 145 (1907).

[5]. Sur les sdries de Dirichlet. Ibid., 148 (1908).

[6]. Sur la sommation des s6ries de Dirichlet. Ibid., 149 (1909).

[7]. Sur les s6ries de Diricblet et les s6ries enti6res. Ibid., 149 (1909).

[8]. Sur u n probl6me d'Abel. Rend. Circ. Mat. Pal., 30 (1910), 339-348.

[9]. U n e mdthode de sommation 6quivalente ~ la mdthode des moyennes arithm6tiques.

C. R. Paris, 152 (1911).

[10]. Uber einen Satz des H c r r n Fatou. J. ]. Math., 140 (1911), 89-99.

[11]. ~ b e r summierbare trigonometrische Reihen. Math. Ann. 71 (1911) 54-75.

[12]. Sur la repr6sentation analytique des fonctions d6finies par des s6ries de Dirichlet. Acta Math., 35 (1911), 253-270.

[13]. Formule d'interpolation pour la d6riv6e d ' u n polynome trigonom6trique. C. R. Paris, 158 (1914).

[14]. Eine trigonometrische Interpolationsformel u n d einige Ungleichungen ffir Polynome.

Jahresber. Deutsch. Math. Ver., 23 (1914), 253-270.

[15]. A survey of the theory of trigonometric series (Swedish). 3 Skand. Mat. Kongr. 1913.

P r i n t e d 1915, 107-127.

[16]. with G. H. ~-~ARDY. The General Theory o/ Dirichlet's Series. Camb. Tracts in Math., 18 (1915), 1-78. Reprinted 1952.

[17]. Neuer Beweis des Fatouschen Satzes. G6tt. Nachr., (1916). 4 pp.

[18]. S~tze fiber Potenzreihen. Ark. Mat. Fys. Astr., l l , no. 12 (1916), 1-16.

[19]. Sur l'hypoth~se de l~iemann. Acta Math., 40 (1916), 185-190.

[20]. U b e r einen Satz des H e r r n Serge Bernstein. Acta Math., 40 (1916), 337-347.

[21]. E i n Konvergenzsatz ffir Diriehletsche Reihen. Acta Math., 40 (1916), 349-361.

[22]. with F. RIESZ. Uber die Randwerte einer analytischen F u n k t i o n . 4. congr, des math. scand.

Stockholm 1916 (1920), 27-44.

[23]. Sur le principe de Phragm6n-Lindel6f. Proc. Camb. Phil. Soc., 20 {1920), 205-207, 21 (1921), 6.

[24]. Sur le probl6me des moments. Premi6re Note. Ark. Mat. Fys. Astr., 16, no. 12 (1921), 1-23. Deuxi6me Note, ibid., 16, no. 19 (1922) 1-21. Troisi6me Note, ibid., 17, no. 16 (1923), 1-52.

[ 25]. Sur la sommation des s6ries de Fourier. Acta Szeged sect. math., 1 ( 1923 ), 104-113.

[26]. Sur u n th6or~me de la moyenne et ses applications. Ibid., 1 (1923), 114-126.

[27]. Sur le probl6me des m o m e n t s et te th6or6me de Parseval correspondent. Ibid., 1 (1923), 209-225.

[28]. Sur l'dquivalence de certaines m6thodes de sommation. Proc. London Math. Soc., (2) 22 (1924), 51-64.

[29]. Sur les fonctions conjugu6es et les s6ries de Fourier. C. R. Paris, 178 (1924).

[30]. ~ b e r die Summierbarkeit durch typische Mittel. Aeta Szeged sect. math., 2 (1924), 51-64.

[31]. with E. HrLB. Neuerc U n t e r s u c h u n g e n fiber trigonometrische Reihen. Enc. d. Math. Wiss., I I C 10 (1924), 1189-1228.

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M A R C E L R I E S Z I N M E M O R I A M x I

[32]. Sur les m a x i m a des formes bilin@aires et sur les fonctioneUes lin@aires. Acta Math., 49 (1927), 465-497.

[33]. Sur les fonctions conjugu@es. Math. Zeitschri]t, 27 (1927), 218-244.

[34]. Sur certaines in@galit@s dans la th@orie des fonctions avec quelques remarques sur les g@om@tries non-euelidiennes. Proe. Roy. P hysiogr. Soc. Lund, 1 (1931), 1-21.

[35]. Sur les ensembles compacts de fonctions sommables. Acta Szeged sect. math., 6, 2-3 (1933), 136-142.

[36]. Z u m Eindeutigkeitssatz der fastperiodischen F u n k t i o n e n . Comm. sdm. math. Lund, 1 (1933), 1-9.

[37]. Eine Bemerkung fiber den Eindeutigkeitssatz der Theorie der fastperiodischen F u n k t i o - hen. Mat. Tidskr. B, 1 (1934), 11-13.

[38]. Int@grale de R i e m a n n - L i o u v i l l e et solution i n v a r i a n t i v e du probl@me de Cauchy pour l'@quation des ondes. C. R. Congres Int. des math. Oslo (1937).

[39]. Modules r@ciproques. Ibid.

[40]. Potentiels de divers ordres et leurs fonctions de Green. Ibid.

[41]. Volumes mixtes et facteurs i n v a r i a n t s dans la th@orie des modules. Ibid.

[42]. Int@grales de R i e m a n n - L i o u v i l l e et potentiels. Acta Szeged sect. math., 9 (1937-38), 1-42.

Rectification .... ibid., 116-118.

[43]. L'int@grale de Riemann-Liouvflle et le probl~me de Cauchy pour l'@quation des ondes.

Soc. math. de France. Con]. ~ la rdunion int. des math. Paris, Juillet, 1937, 1-18.

[44]. A n intuitive picture of non-euclidean geometry. Geometric excursions into relativity theory (Swedish). Aeta Physiogr. Soc. Lund., N F 2, vol. 38, no. 9 (1943), 1-14.

[45]. Sur certaines notions fondamentales en th@orie q u a n t i q u e relativiste. 10. congres de math.

scand. Copenhagen, 1946.

[46]. El@ments de probabflit@ en th@orie q u a n t i q u e relativiste. FSrsdkr. mat. studier till. F i l i p Lundberg. Stockholm, 1946.

[47]. L'int@grale de R i e m a n n - L i o u v i l l e et le probl~me de Cauchy. Acta Math., 81 (1949), 1-223.

[48]. Remarque sur les fonctions holomorphes. Acta Szeged. sect. math., 12 (1950), 53-56.

[49]. Sur le potentiel de Li@nard-Wiechert attach@ ~ u n e ligne d'univers. C. R. Paris, 234 (1952), 2159-2161.

[50]. Sur le potentiel retard@ attach@ k u n courant continu. Ibid., 2260-2261.

[51]. Court expos6 des propri@t@s prineipales de la mesure de Lebesgue. A n n . soc. pol. math., 25 (1952), 298-308.

[52]. Sur le lemme de Zolotareff et sur la loi de r@ciprocit@ des testes quadratiques. Math.

Scand., 1 (1952), 156-169.

[53]. L'dquation de Dirac en relativit@ g@n@rale. 12. congr, des math. seand. Lund, 1954, 241-259.

[54]. with A. E. LlVt~OSTO~E. A short proof of a classical theorem in the theory of Fourier integrals. Amer. Math. Monthly, 62 (1955), 434-437.

[55]. Problems related to characteristic surfaces. Proc. con/. dill. eq. ded. to A. Weinstein. Univ.

of Maryland (1956), 57-71.

[56]. A special characteristic sur]ace--a new relativistic model ]or a particle? F u n d . Res. in Appl.

Math. Dept. Math. Univ. of Maryland. I n t e r i m technical report 25 (1957).

[57]. Cli]]ord numbers and spinets, ch. 1-4. Lecture notes, Inst. Fluid Dyn. Appl. Math. Univ. of Maryland, 1957-58.

[58]. A geometric solution of the wave equation in space-time of even dimension. Comm. Pure A p p l . Math., 13 (1960), 329-351.

[59]. The analytical c o n t i n u a t i o n of the R i e m a n n - L i o u v i l l e integral in the hyperbolic case.

Canad. J. Math., 13 (1961), 37:48.

MARCEL RIESZ IN MEMORIAM