MATEMATIQKI VESNIK

JOVAN KARAMATA

Vo jislav Maric

The great mathematician one of the inter

nationally b est known Serbian scientists Jovan

Karamata was b orn into a rich merchants fa

mily Living in the city of on the bor

ders of the Austrian and Turkish empires and

Serbia several generations of the family develo

p ed a ourishing business through almost three

centuries Even in they acquired an

century baro que mansion in which eight gene

rations of the family have b een living They

b elonged to the upp er class so that even the

Austrian emp eror Joseph the nd when visiting

Zemun in the time of the siege of Belgrade in the AustrianTurkish war

stayed in their home This has b een witnessed by a huge Austrian coat of arms

painted across the ceiling of his ro om bythewell known artist Dimitrije Bacevic

Here the story ab out many known even famous Europ ean families rep eats

the ancestors were soldiers merchants landowners etc usually scientists authors

artists etc J Karamata was b orn in on February but sp entthe

ma jor part of his life until in Zemun There he completed the rst three

classes of high scho ol and then was send by his father b ecause of the First World

War to Up on graduating in he returned to Belgrade enrolled the

Faculty of Engineering of Belgrade University but in following his interest

and talent he moved to the Faculty of PhilosophyDepartment of Mathematics

where he graduated in At the b eginning he was inuenced by his professor

Miha jlo Petrovic the famous Serbian mathematician who himself was a studentof

celebrated French mathematicians Ch Hermite E Picard H Poincare and trans

ferred from to his students in Belgrade imp ortant new mathematical ideas

and theories

Moreover with his colleagues M Milankovic and B Gavrilovic he created an

inspiring atmosphere fruitful for creativework But Karamata considered himself

as a selftaught man acquiring his education by pursuing some b o oks and research

pap ers by masters already at his early studentdays An outstanding p osition in

this resp ect o ccupies the famous b o ok byGPolya and G Szego Aufgab en und

V Maric

Lehrsatze aus der Analysis As a result three months after the graduation he su

bmitted his do ctoral thesis O jedno j vrsti granica slicnih o dredjenim integralima

and obtained his PhD degree already by March

Then he had an usual but fast academic career disturb ed only bythewar

In a Ro ckefeller Foundation fellowinParis in teaching assistant

at the Faculty of Philosophy in assistant professor in

asso ciate one and in the full professor In he b ecame a full professor

at the University of

Karamata however b eing by family p osition deeply ro oted to his country

left it reluctantly nancial motives if any were at the tail of his decision By

the communist regime he was considered as a reactionary b ourgeois capitalist

not friendly to the so cialism although pretending otherwise That minimized

his professional inuence at the Department and made his life frustrating so he

decided to leave But he never broke ties with his country He frequently visited

the Academy lectured at the Mathematical Institute the UniversityofNovi Sad

and the Faculty of Economics in Belgrade and co op erated with his former students

In addition claim to his early international reputation he was elected in

Corresp onding Member of the Yugoslav AcademyofSciences and Arts Zagreb

and in a Corresp onding Member of Regia So cietas Scientiarum Bohemica

Prague In he b ecame a Corresp onding Memb er and in a Full Member

of the Serbian Academy of Sciences and Arts

He was one of the founders of the Mathematical Institute of Serbian Academy

of Sciences and Arts in and the rst EditorinChief of b oth of the journals

Matematicki Vesnik and Publications de lInstitut Mathematique

In he b ecame a memb er of the Editorial Board and Director of the journal

LEnseignementMathematique from Geneva His inuence to the form and content

of this journal was signicant

In he married Emilija Nikola jevic who gave birth to their twosonsand

a daughter His wife died in

After a long illness Jovan Karamata died on August in Geneva His

ashes rest in Zemun

The work of Jovan Karamata b elongs to the classical analysis whichheknew

to p erfection in the time of its prime He had the characteristics and the p ower of

the great analysts of that ep o ch By the words of M Tomic he not only shared

the opinion of GH Hardy that the mathematics was young p eoples game but

demonstrated it Indeed the three most famous of Karamatas pap ers b elong to

the time of his youth and attract more attention to day They app eared in

These three results entered several standard monographs on the sub ject

even some textb o oks and we chose to present these briey among many others

app earing in his scientic pap ers see Bibliography in

Jovan Karamata

New metho d in Taub erian theory

We recall classical N Ab els direct theorem

 

P P

Put s s then s s n implies a x s x the

n n

 

latter series being convergent for jxj

Ie the convergence of the series implies Ab el summability The inverse sta



P

 

as x ie x tement is in general false For example

x 



P

the series is Ab el summable but it obviously diverges But the inverse

statement holds with the additional condition na n as proved

n

byATaub er in Feeling something imp ortant Hardy and Littlewo o d named

such inverse results Taub erian theorems Taub ers pro of was quite elementary

but in JE Littlewood proved the following theorem whichisnow classical

P

Let the series a beAbel summable to a sum s and jna jM thenit

n

converges to s

The generalization of convergence condition to to a nonsp ecialists do es

not seem to b e signicant But the imp ortance of Taub erian theorems emanated

from this theorem of Littlewo o d which b ecame the starting p oint of a new branchof

mathematical analysisthe Taub erian theory His pro of however was extremely

involved and long In spite of the attempts of a number of reputed analysts no

signicant simplications were made Then Karamata published his sp ectacular

pro of in two pages Ub er die HardyLittlewoodschen Umkehrung des Ab elschen

Stetigkeitssatzes Math Zeit

It runs as follows

P

Karamata rst noticed that Ab el summability of the series a to the sum s

implies that for anypower g x x

Z





X

x s g x x s g t dt x





This is then also true for every p olynomial P t Then by applying Weierstrass

approximation theorem Karamata concludes that holds for any bounded R

n

integrable function g x By cho osing x e g x x for e x

n

P

s n s n of and zero otherwise the C summability ie that



the series follows Hence by a simple Hardys lemma its convergence due to

Much later in N Wielandt shared that the C summability step that is the

use of Hardys lemma can b e avoided On the other hand in V Maricand

M Tomic proved that lemma in a couple of lines

The pro of was immediately internationally recognized R Schmidt call it an

imp ortantdiscovery K Knopp said that it was a surprising pro of N Wiener

V Maric

named it elegant EC Titchmarsh extremely elegant etc In on the

o ccasion of the th anniversary of Mathematische Zeitschrift Karamatas pap er

was chosen among the rst most imp ortant ones selected among all of these

published in this journal in that p erio d

Nowadays the pro of can be found in the wellknown treatises of Knopp G

Do etsch Titchmarsh Hardy DV Widder and J Favard

The following true story is nice When visiting the St Andrews universityin

Scotland I was intro duced to ET Copson the author of widely used textb o ok on

the theory of complex functions He told me I haveknown only one Yugoslavian

mathematicianKaramata When co op erating with Hardy at Oxford I found him

one day nervously pacing up and down his oce Not resp onding to greetings he

abruptly said I just received a letter from a young man in Belgrade who claims he

can prove HardyLittlewo o d theorem in just two pages This is simply imp ossible

To understand prop erly why such an excitement ab out a pro of one has to

have in mind that at that time Taub erian theory was one of the highlights of the

analysis Many famous analysts worked on that sub ject Among others Hardy

Littlewo o d in England E Landau F Hausdor Knopp in M Riesz L

Fejer in Hungary etc The signicance and the depth of the theory is best seen

from the fact that S Ikeharas Taub erian theorem contains the celebrated

prime numb ers one x x ln xasx where xisthenumber of primes

not exceeding x

The theory culminated in by Wieners general Taub erian theorem one of

the keystones of Mathematical Analysis to b e quoted b elow It contains Littlewo

o ds theorem as a sp ecial case Yet Karamatas pro of has kept its imp ortance for

at least two reasons

Firstlytoderive Littlewo o ds theorem from the Wieners one is by no means

easy Direct pro of is much more convenient Secondly and more imp ortant the

pro of oers a new metho d which is then used eg for estimating the remainder

R s sasitwas done byseveral authors eg AG Postnikov G Frend J

n n

Korevaar

The theory of slowly varying functions

This app ears to b e the most imp ortant achievementof Karamata The idea

originated in many attempts of relaxing the convergence conditions such as

and The most imp ortantofwhich in Wieners opinion were the ones byR

Schmidt He was the rst to consider a group of terms of a sequence not only a

single one as in and Thus he intro duced slowly oscillating sequences s

n

under the name sehr langsam oscillierende Folgen by requiring s s

 q

with p q and qp

Analogously the functions satisfying f y f x when y x x

and yx or y x are called slowly oscillating The imp ortance of this

class one can recognize from the HR Pitts form of the mentioned Wieners general

Taub erian theorem which reads as follows

Jovan Karamata



Let g L and its Fourier transform gu be dierent from zero for

R R

every u If g x y ay dy l g y dy as x then ay l as y

providedthatay is slow ly oscil lating

Nob o dy b efore Karamata studied these functions as a distinctive class leaving

aside convergence condition This led him to the prop er and very simple denition

A function Lx dened for x is cal led slow ly varying if it is positive

continuous and such that

LtxLx as x for every xed t

Also the function r of the form r xx Lx is cal ledregularly varying is the

index of regular variation

It is Karamatas remarkable achievementtoderive from this denition a whole

theory covering a numb er of useful prop erties of Lfunctions Wemention two most

imp ortant of these The rst one gives the canonical representation

Z

x

Lxcx exp uu

u

a

where cx candx asx

The second one states that the limit holds uniformly in t a b a

b There are not many theories to day which in spite of b eing of a general

reach stem from such a simple denition as It is interesting that his seminal

pap er app eared in an rather unknown journal at that time Sur une mo de de

croissance regulier des fonctions Mathematica Cluj

After Karamata made the rst striking use of slowly varying functions to Ta

ub erian theorems as it will b e shown b elow it was so on realized bya number of

authors that these have various applications in several branches of analysis such

as Ab elian theorems ie asymptotic b ehaviour of integrals Mercerian theorems

complex functions in particular entire ones number theory etc But a real

surprise came by the famous W Fellers b o ok on probability theory in whe

re the great potential of slowly varying functions in that eld b ecame apparent

That was further emphasized byLdeHaanswork in A curious fact is that

Karamata himself was not knowledgeable in probability theory More recentlythe

applications are made to the generalized functions and to dierential equations

b oth linear and nonlinear There are three monographs dealing with the theory

and applications of slowly varying functions by E Seneta in byJGeluk

and de Haan in and the most comprehensiveoneby NH Bingham CM

Goldie and JL Teugels in

Taub erian theorem

In Neuer Beweis und Verallgemeinerung der Taub erschen Satze welche

die Laplacesche und Stietjessche Transformation b etreen Journal fur die reine

V Maric

und angewandte Mathematik Karamata proved the following

result

Let U x be a nondecreasing function in such that

Z



xs

e dfU sg w x



is bounded for every x and if Lx is slow ly varying and w x



x Lx as x then U x x Lx x Analogously

Lx can bereplacedbyLx and x by x

It is the third imp ortant discovery of Karamata that we mention which is

to dayknown as Karamata Taub erian theorem or sometimes as HardyLittlewood

Karamata one



In the original statementonlypower x or x app eared ie L Hardy

and Littlewo o d realized very early that a factor of slower increase could b e intro

n

Q

duced so they put log xt where log x stands for the k th iteration of the

k

k k



logarithm and t is a real numb er Then the natural question arises what kind

k

of factors one could use and also should all of these require a new pro of Then

Karamata used L as a factor obtaining thus the most general form of the theo

rem and with an elegant pro of Seneta considers this to b e the most famous and

very widely useful theorem in probability theory and Bingham even as one of the

ma jor analytic results of this century

Another ma jor Karamatas contribution to mathematics was in educating yo

ung mathematicians He was a devoted teacher and published several textb o oks

some of which b eing quite original More imp ortant he had a rare abilitytodis

cover and promote talented students to start research at an early stage and then

lead them to the scientic maturity To appraise his merits prop erly one has to

knowhow dicult was the situation in our country immediately after the Second

World War in many resp ect It was undergoing the so called revolutionary chan

ges which included p olitical pressure even p ersecution of a number of students

and professors On the other hand the well equipp ed library of the Departmentof

Mathematicsthe only one of the kind was destroyed bythewar b ombing In spite

of all that Karamata b esides of teaching worked with his students in the Academy

of Science in Hotel Ma jestic at his home He landed them books from his own

library For those who were considered as p olitically unreliable even enemies he

was their only hop e Later in Geneva he help ed a numb er of them to get foreign

scholarship for further studies and p ositions at foreign universities He continued

to co op erate and to write joint pap ers with some of them to the end of his life

Thus he created what was later named by Seneta and Bingham as the Yugoslav

scho ol of analysis but that was in fact Karamatas scho ol

We conclude with the following additional remarks Karamata wrote his pa

p ers with an utmost care although he was able to write several pages of awless

calculations in no time and in a very legible and harmonious handwriting at that

Jovan Karamata

On the other hand once b eing ill he outlined an entire pro of without writing a

single character Ein Konvergentzsatz fur trigonometrische Integrale Jour

nal fur die reine und angewandte Mathematik His pap ers are

characterized by clarity and elegance

Yet in everyday life he showed no interest in any kind of art nor was know

ledgeable in the literature

On the other hand he exhibited a somewhat surprising interest in antrop o

sophical do ctrine of Rudolf Steinera mixture of philosophy and mysticism He

enjoyed talking ab out the sub ject and it was very interesting how he puried so

me rather nebulous statements of the do ctrine through his p owerful logic to make

these almost plausible

REFERENCES

 GH Hardy Divergent Series Oxford

 E Seneta Regularly Varying Functions Lecture Notes in Mathematics Springer Berlin

Heidelb erg NewYork

 JL Geluk L de Haan Regular Variation Extensions and Tauberian Theorems CWI Tract

Amsterdam

 NN Bingham CM Goldie and JL Teugels Regular Variation Encyclopaedia of Mathema

tics and its ApplicationsVol Cambridge University Press Cambridge

 M Tomic S Aljancic Remembering Jovan Karamata Publ Inst Math 

 A Nikolic Jovan Karamata zivot kroz matematikuZaduzbina Andrejevic Beograd

 M Tomic Jovan Karamata Bulletin T CXXI I de lAcademie Serb e des Sciences et des Arts Classe Sci Math Not Sciences Math