In the history of mathematics there are quite a few persons
whom we prefer to recollect in pairs. Listed among
them are Euclid and Diophant, Newton and Leibniz, Bolyai and Lobachevskii, Hilbert
and Poincaré, as well as Bourbaki
and Arnold. In this series we enroll Sobolev and
Schwartz who are inseparable from one of the most brilliant discoveries of the
twentieth century, the theory of generalized functions or distribution theory,
providing a revolutionary new approach to partial differential equations.
The most vibrant and lasting achievements of mathematics
reside in formulas and lists. There are pivotal distinctions between lists and
formulas. The former deposit that which was open for us.
The lists of platonic solids, elementary catastrophes, and finite simple groups
are next of kin to the Almagest and herbaria. They are the objects of
admiration, tremendous and awestruck. The article of the craft of mathematics
is a formula. Each formula enters into life as an instance of
materialization of mathematical creativity. No formula serves only the purpose
it was intended to. In part, any formula is reminiscent of household
appliances, toys, or software. It is a very rare event that somebody reads the
user's guide of a new TV set or the manual for running a new computer program.
Usually everyone utilizes his or her new gadgets experimentally by pressing
whatever keys and switches. In much the same way we handle formulas. We
painstakingly “twist and turn” them, audaciously insert new parameters,
willfully interpret symbols, and so on.
Mathematics is the craft of formulas and the art of
calculus. If
someone considers this claim as feeble and incomplete, to remind is in order
that, logically speaking, set theory is just an instance of the first order
predicate calculus.
Distribution theory has become the calculus of today. Of
such a scale and scope is the scientific discovery by Sobolev
and Schwartz.
Sobolev was born in St. Petersburg
on October 6, 1908 in the family of Lev Aleksandrovich
Sobolev, a solicitor. Sobolev's
grandfather on his father's side descended from a family of Siberian
Cossacks.
Sobolev was bereaved of his father
in the early childhood and was raised by his mother Natal'ya
Georgievna who was a highlyeducated teacher of
literature and history. His mother also had the second speciality:
she graduated from a medical institute and worked as a tutor at the First
Leningrad Medical Institute. She cultivated in Sobolev
the decency, indefatigability, and endurance that characterized him as a
scholar and personality.
Sobolev fulfilled the program of
secondary school at home, revealing his great attraction to mathematics. During
the Civil War he and his mother lived in Kharkov. When living there, he studied
at the preparatory courses of a evening technical
school for one semester. At the age of 15 he completed the obligatory programs
of secondary school in mathematics, physics, chemistry, and other natural
sciences, read the classical pieces of the Russian and world literature as well
as many books on philosophy, medicine, and biology.
After the family had transferred from Kharkov to Petersburg
in 1923, Sobolev entered the graduate class of School
No. 190 and finished with honors in 1924, continuing his study at the
First State Art School in the piano class. At the same year he entered the
Faculty of Physics and Mathematics of Leningrad State University (LSU) and
attended the lectures of Professors N. M. Günter,
V. I. Smirnov, G. M. Fikhtengolts,
and others. He made his diploma on the analytic solutions of a system of
differential equations with two independent variables under the supervision of
Günter.
Günter propounded the idea that the set functions are
inevitable in abstracting the concept of solution to a differential equation.
Günter's approach influenced the further train of thought of Sobolev.^{2}
After graduation from LSU in 1929, Sobolev
started his work at the Theoretical Department of the Leningrad Seismological
Institute. In a close cooperation with Smirnov he then solved some fundamental
problems of wave propagation. It was Smirnov whom Sobolev
called his teacher alongside Günter up to his terminal days.
Since 1932 Sobolev worked at the Steklov Mathematical Institute in Leningrad; and since
1934, in Moscow. He continued the study of hyperbolic equations and proposed a
new method for solving the Cauchy problem for a hyperbolic equation with
variable coefficients. This method was based on a generalization of the
Kirchhoff formula.
Research into hyperbolic equations led Sobolev
to revising the classical concept of a solution to a differential equation. The
concept of a generalized or weak solution of a differential equation was
considered earlier. However, it was exactly in the works by Sobolev
that this concept was elaborated and applied systematically. Sobolev posed and solved the Cauchy problem in spaces of functionals, which was based on the revolutionary extension
of the Eulerian concept of function and declared 1935
as the date of the birth of the theory of distributions.
Suggesting his definition of generalized derivative, Sobolev enriched mathematics with the spaces of functions
whose weak derivatives are integrable to some power.
These are now called Sobolev spaces.
Let f and g be locally summable
functions on an open subset G of _{}, and let a be a
multiindex. A function g, denoted by _{}, is the generalized derivative in the Sobolev sense or
weak derivative of f of order _{} provided that

for every test
function_{}, i. e. such that the support
of _{} is a compact subset of G and j is _{} times continuously
differentiable in G, where _{} is the
classical derivative of _{} of
order _{}. The vector space W^{l}_{p},
with _{}, of the (cosets of) locally summable f on G whose all weak derivatives _{} with _{} are pintegrable in G becomes a Banach space under the norm:

Sobolev found the general criteria
for equivalence of various norms on W_{p}^{l}
and showed that these spaces are the natural environment for posing the
boundary value problems for elliptic equations. This conclusion was based on
his thorough study of the properties of Sobolev
spaces. The most important facts are embedding theorems. Each embedding
theorem estimates the operator norm of an embedding, yielding
special inequalities between the norms of one and the same function
inside various spaces.
The contributions of Sobolev
brought him recognition in the USSR. In 1933 Sobolev
was elected a corresponding member of the Academy of Sciences at the age of 24
years. In 1939 he became a full member of the Academy and remained the youngest
academician for many years.
Inspired by military
applications in the 1940s, Sobolev studying the
system of differential equations describing small oscillations of a rotating
fluid. He
obtained the conditions for stability of a rotating body with a filledin
cavity which depend on the shape and parameters of the cavity. Moreover, he
elaborated the cases in which the cavity is a cylinder
or an ellipsoid of rotation. This research by Sobolev
signposted another area of the general theory which concerns the Cauchy and
boundary value problems for the equations and systems that are not solved with
respect to higher time derivatives.
In the grievous years of the Second World War from 1941 to
1944 Sobolev occupied the position of the director of
the Steklov Mathematical Institute.
Sobolev was one of the first
scientists who foresaw the future of computational mathematics and cybernetics.
From 1952 to 1960 he held the chair of the first national department of
computational mathematics at Moscow State University. This department has
played a key role in the development of this important area of the today's
mathematics.
Addressing the problems of computational mathematics, Sobolev lavishly applied the apparatus of the modern
sections of the theoretical core of mathematics. It is typical for him to pose
the problems of computational mathematics within functional analysis. Winged
are his words that “to conceive the theory of computations without Banach
spaces is impossible just as trying to conceive it without computers.”
It is worthwhile to emphasize the great role in the uprise of cybernetics and other new areas of research in
this country which was played by the publications and speeches of Sobolev who valiantly defended the new trends in science
from the ideologized obscurantism.
To overrate is diffucult the contribution of Sobolev to the design of the nuclear shield of this
country. From the first stages of the atomic project of the USSR he was listed
among the top officials of Laboratory No. 2 which was renamed for secrecy
reasons into the Laboratory of Measuring Instruments (abbreviated as LIPAN in
Russian). Now LIPAN lives as the Kurchatov Center.
The main task of the joint work with I. K. Kikoin
was the implementation of gaseous diffusive uranium enrichment for creation of
a nuclear explosive device.
Sobolev administered and supervised
various computational teams, studied the control of the industrial processes of
isotope separation, struggled for the low costs of production and made
decisions on many managerial and technological matters. For his contribution to
the Abomb project, Sobolev twice gained a Stalin
Prize of the First Degree. In January of 1952 Sobolev
was awarded with the highest title of the USSR: he was declared the Hero of the
Socialist Labor for exceptional service to the state.
Sobolev's research was inseparable
from his management in science. At the end of the 1950s M. A. Lavrent'ev, S. L. Sobolev,
and S. A. Khristianovich came out with the
initiative to organize a new big scientific center, the Siberian Division of
the Academy of Sciences. For many scientists of the first enrolment to the
Siberian Division it was the example of Sobolev, his
authority in science, and the attraction of his personality that yielded the
final argument in deciding to move to Novosibirsk.
The Siberian period of Sobolev's
life in science was marked by the great achievements in the theory of cubature
formulas. Approximate integration is one of the main problems in the theory of
computations—the cost of computation of multidimensional integrals is
extremely high. Optimizing the formulas of integration is understood now to be
the problem of minimizing the norm of the error on some function space. Sobolev suggested new approaches to the problem and
discovered marvelous classes of optimal cubature formulas.
Sobolev merits brought him many
decorations and signs of distinction. In 1988 he was awarded the highest prize
of the Russian Academy of Sciences, the Lomonosov
Gold Medal.
Sobolev passed away in Moscow on
January 3, 1989.
Schwartz was born in Paris on March 5, 1915 in the family of
Anselme Schwartz, a surgeon. There
were quite a few prominent persons among his next of kin. J. Hadamard was his granduncle. Many celebrities are listed in
the lone of his mother's line Claire Debrés:
Several Gaullist politicians belonged to the Debérs.
In 1938 Schwartz married MarieHélèn
Lévy, the daughter of the outstanding
mathematician P. Lévy who was one of the
forefathers of functional analysis. MarieHélèn
had become a professional mathematician and gained the position of a full
professor in 1963.
The munificent gift of Schwartz was revealed in his lecée years. He won the most prestigious competition
for high school students, Concours Général in Latin. Schwartz was unsure about
his future career, hovering between geometry and “classics” (Greek and Latin).
It is curious that Hadamard had a low opinion of
Schwartz mathematical plans, since the sixteenyears old Laurent did not know
the Riemann zeta function. By a startling contrast, Schwartz
was boosted to geometry by the pediatrician Robert Debré
and one of his teachers of classics.
In 1934 Schwartz passed examinations to the Ècole Normale Supérieure
(ENS) after two years of preparation. He was admitted together with Gustave Choquet, a winner of
the Concours Général
in mathematics, and MarieHélčn, one of the
first females in the ENS. The mathematical atmosphere of those years in the ENS
was determined by Č. Borel, Č. Cartan, A. Denjoy, M. Fréchet, and P. Montel.
The staff of the neighboring Collčge de France
included H. Lebesgue who delivered lectures and Hadamard who conducted seminars. It was in his student
years that Schwartz had acquired his irretrievable and permanent love to
probability theory which grew from conversations with his future fatherinlaw Lévy.
After graduation from the ENS
Schwartz decided to be drafted in the compulsory military service for two
years. He
had to stay in the army in 1939–1940 in view of the war times. These years were
especially hard for the young couple of the Schwartzes.
It was unreasonable for Jews to stay in the occupied zone. The Schwartzes had to escape from the native north and manage
to survive on some modest financial support that was offered in particular by
Michelin, a worldrenowned tire company. In 1941 Schwartz was in Toulouse for a
short time and met H. Cartan and J. Delsarte who suggested that the young couple should move to
ClermontFerrand, the place of temporary residence of the group of professors
of Strasbourg University which had migrated from the German occupation. These
were J. Dieudonné, Ch. Ehresmann, A. Lichnerowicz,
and S. Mandelbrojt. In ClermontFerrand Schwartz
completed his Ph. D. Thesis on approximation of a real function on the
axis by sums of exponentials.
Unfortunately, the war had intervened into the mathematical
fate of Schwartz. His family had to change places under false identities.
Curiously, in the time of the invention of distributions in November of 1944
Schwartz used the identity of Selimartin. The basics
of Schwartz's theory appeared in the Annales
of the University of Grenoble in 1945 [8]. Schwartz described the process of invention as
“cerebral percolation.” After a year's stay in Grenoble, Schwartz acquired a
position in Nansy, plunging to the center of “Bourbakism.” It is well known that N. Bourbaki resided in Nancago,
a mixture of Nancy and Chicago. A. Weyl
lived in Chicago, while Delsarte and Dieudonné were in Nancy. Before long
Schwartz was enrolled in the group of Bourbaki.
In 1950 he was awarded with the Fields medal for distribution theory. His
nowcelebrated twovolume set Théorie
des Distributiones was printed a short time
later.
In 1952 Schwartz
returned back to Paris and began lecturing in Sorbonne; and since 1959, in the Čcole Politechnique in company
with his fatherinlaw Lévy. Many celebrities
were the direct students of Schwartz. Among them we list A. Grothendieck, J.L. Lions,
B. Malgrange, and A. Martineau.
Schwartz wrote: “To discover
something in mathematics is to overcome an inhibition and a tradition. You
cannot move forward if you are not subversive.” This statement is in good
agreement with the very active and versatile public life of Schwartz. He joined
Trotskists in his green years, protesting against the
monstrosities of capitalism and Stalin's terror of the 1930s. Since then he had
never agreed with anything that he viewed as violation of human rights,
oppression, or injustice. He was very active in struggling against the American
war in Vietnam and the Soviet invasion in Afghanistan. He fought for liberation
of a few mathematicians that were persecuted for political reasons, among them
Jose Luis Massera, Vaclav Benda, et al.
Schwartz was an outstanding
lepidopterist and had collected more than 20 000 butterflies. It is not by
chance that the butterflies are depicted on the soft covers of the second
edition of his Théorie des Distributiones.
Schwartz passed away in Paris
on June 4, 2002.
Distribution theory stems
from the intention to apply the technologies of functional analysis to studying
partial differential equations. Functional analysis rests on algebraization, geometrization,
and socialization of analytical problems. By socialization we usually mean the
inclusion of a particular problem in an appropriate class of its congeners.
Socialization enables us to erase the “random features,”^{3} eliminating the difficulties of the insurmountable
specifics of a particular problem. In the early 1930s the merits of functional
analysis were already demonstrated in the area of integral equations. The time
was ripe for the differential equations to be placed on the agenda.
It is worth observing that
the contemplations about the nature of integration and differentiation underlie
most of the theories of the presentday functional analysis. This is no wonder
at all in view of the key roles of these remarkable linear operations. Everyone
knows that integration possesses a few more attractive features than
differentiation: the integral is monotone and raises smoothness. n
Derivation lacks these nice properties completely. Everyone
knows as well that the classical derivative yields a closed yet unbounded
operator (with respect to the natural uniform convergence topology that is
induced by the Chebyshev supnorm). The series of
smooth functions cannot be differentiated termwise in
general, which diminishes the scope of applications of analysis to differential
equations.
There is practically no
denying today that the concept of generalized derivative occupies a central
place in distribution theory. Derivation is now treated as the operator that
acts on the nonsmooth functions according to the same
integral laws as the procedure of taking the classical derivative. It is
exactly this approach that was pursued steadily by Sobolev.
The new turnpike led to the stock of previously impossible differentiation
formulas. It turned out that each distribution possesses derivatives of all
orders, every series of distributions may be differentiated termwise
however often, and many “traditionally divergent” Fourier series admit
presentations by explicit formulas. Mathematics has acquired additional
fantastic degrees of freedom, which makes immortal the name of Sobolev as a pioneer of the calculus of the twentieth
century.
The detailed expositions of
the new theory by Sobolev and Schwartz had appeared
practically at the same time. In 1950 the first volume of Théorie des Distributiones
was published in Paris, while Sobolev's book Applications
of Functional Analysis in Mathematical Physics was printed in Leningrad.
In 1962 the Siberian Division of the Academy of Sciences of the USSR
reprinted the book, while in 1963 it was translated into English by the
American Mathematical Society. The second edition of the Schwartz book was
published in 1966, slightly enriched with a generalized version of the de Rham currents. Curiously, Schwartz left the historical
overview practically the same as in the introduction to the first edition.
The new methods of
distribution theory turned out so powerful as to enable mathematicians to write
down, in explicit form, the general solution of an arbitrary partial
differential equation with constant coefficients. In fact, everything reduces
to existence of fundamental solutions; i. e. to
the case of the Dirac deltafunction on the righthand side of the equation
under consideration. The existence of these solutions was already established in 1953
and 1954 by B. Malgrange and L. Ehrenpreis independently of each other. However, it was
only in 1994 that some formula for a fundamental solution was exhibited by
H. König. Somewhat later N. Ortner and P. Wagner found a more elementary formula.
Their main result is as follows:
Theorem. Assume that _{}, where P is
a polynomial of degree m. Assume further that _{} and _{}, where P_{m}
is the principal part of, P. Then the distribution E given as

is a fundamental solution of the operator _{}. Moreover, _{}.
It stands to reason to inspect the structure of the formula
which reveals the role of the distributional Fourier transform _{} and the Schwartz space _{} comprising tempered distributions.^{5}
The
existence of a fundamental solution of an arbitrary partial differential equation
with constant coefficients is reverently called the Malgrange–Ehrenpreis
Theorem. It is hard to overestimate this splendid achievement which remains
one of the splendid triumphs of the abstract theory of topological vector
spaces.
The
road from solutions in distributions to standard solutions lies through Sobolev spaces. Study of the embeddings and traces of Sobolev spaces has become one of the main sections of the
modern theory of real functions. Suffice it to mention S. M. Nikol'skii, O. V. Besov,
G. Weiss, V. P. Il'in, and V. G. Mazya in order to conceive the greatness of this area of
mathematical research. The titles of dozen books mention Sobolev
spaces, which is far from typical in the presentday science.
The
broad stratum of modern studies deals with applications of distributions in
mathematical and theoretical physics, complex analysis, the theory of pseudodifferential operators, Tauberian
theorems, and other sections of mathematics.
The
physical sources of distribution theory as well as the ties of the latter with
theoretical physics, are the topics of paramount
importance. They require a special scrutiny that falls beyond the scope of this
article.^{5} We will confine exposition to the
concise historical comments
by V. S. Vladimirov:^{6}
It was already the creators of this
theory, S. L. Sobolev [5] and
L. Schwartz [19] who
studied the applications of the theory of generalized functions in mathematical
physics. After a conversation with S. L. Sobolev
about generalized functions, N. N. Bogolyubov
used the Sobolev classes [3] of test
and generalized functions C^{m}_{comp} and (C^{m}_{comp})^{*}
in constructing his axiomatic quantum field theory [20]–[22]. The same related to the Wightman axiomatics [23]. Moreover,
it is impossible in principle to construct any axiomatics
of quantum field theory without generalized functions. Furthermore, in the
theory of the dispersion relations [24] that are derived from the Bogolyubov
axiomatics, the generalized functions, as well as
their generalizationshyperfunctions, appear as the
boundary values of holomorphic functions of (many)
complex variables. This fact, together with the relevant aspects such as Bogolyubov's “EdgeoftheWedge” Theorem, essentially
enriches the theory of generalized functions.
J. Leray was one of the most prominent French mathematicians
of the twentieth century. He was awarded with the Lomonosov
Gold Medal together with Sobolev in 1988. Reviewing
the contributions of Sobolev from 1930 to 1955
in the course of Sobolev's election to the Academy of
Sciences of the Institute of France in 1967, J. Leray
wrote:
Distribution
theory is now well developed due to the theory of topological vector spaces and
their duality as well as the concept of tempered distribution which is one of
the important achievements of L. Schwartz (Paris) which enabled him to
construct the beautiful theory of the Fourier transform for distributions;
G. de Rham supplied the concept of distribution
with that of current which comprises the concepts of differential form and
topological chain; L. Hörmander (Lund,
Princeton), B. Malgrange (Paris),
J.L. Lions (Paris) used the theory of distributions to renew the theory of
partial differential equations; while P. Lelong
(Paris) established one of the fundamental properties of analytic sets. The
comprehensive twovolume treatise by L. Schwartz and even more
comprehensive fivevolume treatise by Gelfand and Shilov (Moscow) are the achievements of so great an
importance that even the French contribution deserves the highest awards of our
community. The applications of distribution theory in all areas of mathematics,
theoretical physics, and numerical analysis remind of the dense forest hiding
the tree whose seeds it has grown from. However, we know that if Sobolev had fail to make his discovery about 1935 in
Russia, it would be committed in France by 1950 and somewhat later in Poland;
the USA also flatters itself that they would make this discovery in the same
years: The science and art of mathematics would be late only by 15 years as
compared with Russia....
In
sharp contrast with this appraisal, we cite F. Tréves
who wrote in the memorial article about Schwartz in October 2003 as follows:
Sobolev in his articles [Sobolev, 1936] and [Sobolev,
1938]^{7} (Leray used to refer to
“distributions, invented by my friend Sobolev”). As a
matter of fact, Sobolev truly defines the distributions
of a given, but arbitrary, finite order m: as the continuous linear functionals on the space C^{m}_{comp}
of compactly supported functions of class C^{m}. He keeps the integer m
fixed; he never considers the intersection C^{∞}_{comp} of the spaces C^{m}_{comp} for all m. This is all the
more surprising, since he proves that C^{m+1}_{comp} is dense
in C^{m}_{comp} by the Wiener
procedure of convolving functions f Î
C^{m}_{comp} with a
sequence of functions belonging to C^{∞}_{comp}. In connection with
this apparent blindness to the possible role of mentioned to Henri Cartan his inclination to use the elements of C^{∞}_{comp}
as test functions, Cartan tried to dissuade him:
“They are too freakish (trop monstrueuses).”
Using
transposition, Sobolev defines the multiplication of
the functionals belonging to C^{m}_{comp} by the functions belonging
to C^{m} and the differentiation of those functionals:
d/dx maps (C^{m}_{comp})^{*} into
(C^{m+1}_{comp})^{*}. But again
there is no mention of Dirac δ(x) nor of
convolution, and no link is made with the Fourier transform. He limits himself
to applying his new approach to reformulating and solving the Cauchy problem
for linear hyperbolic equations. And he does not try to build on his remarkable
discoveries. Only after the war does he invent the Sobolev
spaces H^{m} and then only for integers m > 0. Needless
to say, Schwartz had not read Sobolev's articles,
what with military service and a world war (and Western mathematicians.
ignorance of the works of their Soviet colleagues). There is no doubt that
knowing those articles would have spared him months of anxious uncertainty.
F. Tréves should be honored for drifting aside from the
practice of evaluating publications from what they lack when he wrote somewhat
later about that which made the name of Schwartz immortal:
Granted
that Schwartz might have been replaceable as the inventor of distributions,
what can still be regarded as his greatest contributions to their theory? This
writer can mention at least two that will endure: (1) deciding that the
Schwartz space S of rapidly decaying
functions at infinity and its dual S¢ are the
“right” framework for Fourier analysis, (2) the Schwartz kernel theorem.
The
Tréves opinion coincides practically verbatim
with the narration of Schwartz in his autobiography published firstly in 1997.
Moreover, Schwartz had even remarked there about Sobolev
that^{8}
he did not develop his theory in view
of general applications, but with a precise goal: he wanted to define the
generalized solution of a partial differential equation with a second term and
initial conditions. He includes the initial conditions in the second term in
the form of functionals on the boundary and obtains
in this way a remarkable theorem on hyperbolic second order partial
differential equations. Even today this remains one of the most beautiful
applications of the theory of distributions, and he found it in a rigorous
manner. The astounding thing is that he stopped at this point. His 1936
article, written in French, is entitled “Nouvelle méthode
ŕ résoudre de problčme
de Cauchy pour les équations linéares hyperboliques normales.” After this article, he did nothing further in
this fertile direction. In other words, Sobolev
himself did not fully understand the importance of his discovery.
It
is impossible to agree with these opinions. Rather strange is to read about the
absence of any mention of the Dirac deltafunction among the generalized
functions of Sobolev, since d obviously belongs to each of
the spaces (C^{m}_{comp})^{*}.
Disappointing
is the total neglect of the classical treatise of Sobolev [6] which was a deskbook of many specialists in functional analysis and
partial differential equations for decades.^{9}
Finally, Schwartz was not recruited in 1997 and did not participate in a`world war. Therefore, there were some other reasons for
him to overlook the Sobolev book [8]
which contains the principally new appications of distributions to
computational mathematics.
Sobolev based his pioneering results
in numerical integration on developing the theory of the Fourier transform of
distributions which was created by Schwartz.
Prudent
in the appraisals, exceptionally tactful, and modest in his ripe years, Sobolev always abstained from any bit of details of the
history of distribution theory neither in private conversations nor in his
numerous writings. The opinion that he decided worthy to be left to the future
generations about this matter transpires in his concise comments on the origins
of distribution theory in his book [8,Ch. 8]
which was printed in 1974:
The generalized functions are "ideal elements" that complete the
classical function spaces in much the same way as the real numbers complete the
set of rationals.
In this chapter we concisely present the theory of these functions which we
need in the sequel. We will follow the way of presentation close to that which
was firstly used by the author in 1935 in [16].^{10} The theory
of generalized functions was further developed by L. Schwartz [21]
who has in particular considered and studied the Fourier transform of
a generalized function.^{11}
Historically, the generalized function had appeared explicitly in the studies
in theoretical physics as well as in the works of J. Hadamard,
M. Riesz, S. Bochner,
et al.
Therefore, we can agree only
in part with the following statement by Schwartz [11,p. 236]:
Sobolev and I and all the others who came
before us were influenced by our time, our environment and our own previous
work. It makes it less glorious, but since we were both ignorant of the work of
many other people, we still had to develop plenty of originality.
Most
mathematicians agree that Israel Gelfand could be
ranked as the best arbiter in distribution theory. The fivevolume series Generalized
Functions written by him and his students was started in the mid 1950s and
remains one of the heights of the world mathematical literature, the
encyclopedia of distribution theory. In the preface to the first edition of the
first volume of this series, Gelfand wrote:
It
was S. L. Sobolev who introduced
generalized functions in explicit and now generally accepted form
in 1936.... The monograph of Schwartz Théorie
des Distributiones appeared in 19501951. In
this book Schwartz systemized the theory of generalized functions,
interconnected all previous approaches, laid the theory of topological linear
spaces in the foundations of the theory of generalized functions, and obtain a number of essential and farreaching results. After
the publication of Théorie des Distributiones, the generalized functions won
exceptionally swift and wide popularity just in two or three years.
This
is an accurate and just statement. We may agree with it.
Pondering
over the fates of Sobolev and Schwart, it is impossible to obviate the problem of
polarization of the opinions about the mathematical discovery of these scholars.
The hope is naive that this problem will ever received a simple and definitive
answer that satisfies and convinces everyone. It suffices to consider the
available experience that concerns other famous pairs of mathematicians whose
fates and contributions raise the quandaries that sometimes lasted for
centuries and resulted in the fierce clashes of opinions up to the present day.
The sources of these phenomena seem of a rather universal provenance that is
not concealed in the particular personalities but resides most probably in the
nature of mathematical creativity.
Using
quite a risky analogy, we may say that mathematics has some features associated
with the trends of artistry which are referred to traditionally as classicism
and romanticism. It is hard to fail discerning the classic lineaments of the
Hellenistic tradition in the writings of Euclid, Newton, Bolyai, Hilbert, and
Bourbaki. It is impossible to fail to respond to the accords of the romantic
anthem of the human genius which sound in the pages of the writings of
Diophant, Leibniz, Lobachevskii, Poincaré, and Arnold.
The
magnificent examples of mathematical classicism and romanticism glare from the
creative contributions of Sobolev and Schwartz. These giants and their
achievements will remain with us for ever.
Kutateladze S. S. “Sergei Sobolev and Laurent Schwartz,” Herald of the Russian
Academy of Sciences, 74:2, pp. 183188 (2005).
Soboleff S. L., “Le probléme de Cauchy dans l'espace des fonctionnelles,” C. R. Acad. Sci. URSS, 3:7,
pp. 291294 (1935).
Sobolev S. L., “Méthode nouvelle ŕ résoudre le
probléme de Cauchy pour les équations linéaires hyperboliques,” Sbornik,
1, No. 1, pp. 3970 (1936).
Sobolev S. L., “About one theorem of functional analysis,” Sbornik, 4, No. 3, pp. 471496 (1938).
Sobolev S. L., Applications of Functional Analysis in
Mathematical Physics. Leningrad: Leningrad University Press (1950).
Sobolev S. L., Introduction to the Theory of Cubature
Formulas. Moscow: Nauka Publishers (1974).
Blok
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^{1}Partly printed in [1]
with unauthorized omissions.
The author
thanks V. A. Aleksandrov and
V. P. Golubyatnikov who helped him in
better understanding of French sources. The author is especially grateful to
Yu. L. Ershov who was persistent in
inviting the author to make a talk in the special session of the Academic Council
of the Sobolev Institute on October 14, 2003.
The present article bases on this talk. The author acknowledges the subtle and
deep comments of V. I. Arnold and V. S. Vladimirov
on the preprint of a draft of the talk which led to many improvements.
^{2}It was A. M. Vershik and V. I. Arnold who attracted the author's
attention to the especial role of Günter in the prehistory of distribution
theory.
^{3}This is a cliché with
a centuryold history. The famous Russian symbolist Alexander Blok (18801921)
used the concept of random feature in his
incomplete poem “Revenge” as of
1910 [7,p. 482]. The prologue of this
poem contains the lines that are roughly rendered in English as follows:
You share the gift of prudent measure
For what keen vision might perceive.
Erasing random
features, treasure
The world of beauty to receive.
^{4}Also known as “generalized
functions of slow growth.”
^{5}Some historical details are
collected in [26]. Also see the
article [27]. J.M. Kantor kindly made
this article available to the author before publication with a courteous
cooperation of Ch. Davis, EditorinChief of The Mathematical
Intelligencer. It was the proposal of Ch. Davis that the article by
J.M. Kantor be supplemented with the short
comments [28] and [29].
^{6}Cited from the handwritten
review for the Herald of the Russian Academy of Sciences, dated as of
December 10, 2003.
^{7}These are references to the
articles in Sbornik [2, 3].
^{9}Published in 1950 by
Leningrad State University, reprinted in 1962 by the Siberian Division of
the Academy of Sciences of the USSR in Novosibirsk, and translated into
English by the American Mathematical Society in 1963. The third Russian
edition was printed by the Nauka Publishers
in 1988.
^{10}This is a reference to
the article of 1936 in Sbornik [3].
^{11}Cp. [25,
p. 355]). This is a curious misprint: the correct reference to Schwartz's
twovolume set should be [47].
Sobolev Institute
of Mathematics
Koptyug's Avenue 4
Novosibirsk,
630090
RUSSIA
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