Topic #25  -----------------  OP-SF NET  ------------------ May 17, 1995
                             ~~~~~~~~~
From: Richard Askey 
Subject: Gabor Szego - One hundred years  (long topic)

A memorial celebration of the 100-th anniversary of Gabor
Szego's birth (on January 20, 1895 in Kunhegyes, Hungary) was held
on January 21, 1995 in Kunhegyes. See OP-SF Net 2.1, Topic 6 for a short
account.
One of the speeches delivered there
was an essay written by Dick Askey. It was translated into Hungarian
and it was read at the celebration by Gyuri Petruska.
The essay follows below (in English), with
all diacritical (TeX) signs suppressed.


                     GABOR SZEGO - ONE HUNDRED YEARS

                             by Richard Askey

It is 100 years since Gabor Szego was born and 80 years since the
publication of his first paper. Before this publication he had already
shown a strong talent in mathematics by winning the Eotvos competition
in 1912. His first paper contained the solution of a problem of George
Polya. However, he had earlier, in 1913, published the solution of
another problem of Polya. For the nonmathematicians it should be
remarked that there are problems at various levels. Some, like those
you did in school, are ones which everyone should learn how to do.
Then there are contest problems, like those in the Eotvos competition.
These are harder and frequently require deeper insight than seems so
at first reading. The problem of Polya which Szego solved and
published in 1913 is an example of a harder type, which has long
attracted prospective mathematicians. Hungary has long specialized in
the use of problems to attract young students to mathematics, and
other countries have learned from you and have contests of mathematics
problems to encourage students to think harder than is needed for the
typical school problem. The problem of Polya which made up Szego's
first published paper was an open problem and there is still great
interest in extensions of Szego's solution to more complicated
problems of this nature.

Since Szego had been a mathematical prodigy himself, he was an ideal
person to be asked to tutor one of the great mathematical minds of
this century, John von Neumann. Here is what Norman Macrae wrote in
his book "John von Neumann".

"Professor Joseph Kurschak [of the Lutheran School in Budapest] soon
wrote to a university tutor, Gabriel Szego, saying that the Lutheran
School had a young boy of quite extraordinary talent. Would Szego, as
was the Hungarian tradition with infant prodigies, give some
university teaching to the lad?


"Szego's own account of what happened was modest. He wrote that he
went to the von Neumann house once or twice a week, had tea, discussed
set theory, the theory of measurement, and some other subjects with
Jancsi, and set him some problems. Other accounts in Budapest were
more dramatic. Mrs.  Szego recalled that her husband came home with
tears in his eyes from his first encounter with the young prodigy. The
brilliant solutions to the problems posed by Szego, written by Johnny
on the stationery of his father's bank, can still be seen in the von
Neumann archives in Budapest."

Macrae was wrong in saying Kurschak was at the Lutheran School. He was
a professor at the university. He was the appropriate contact between
the mathematics teacher at the Lutheran School and Szego.

Szego served in the army in the First World War, but continued to do
mathematics, and received his Ph.D. in 1918 in Vienna. Fifty years
later he returned to Vienna for a celebration of this, and I still
remember how pleased he was with this when he described it to me a few
years later.  After temporary positions in Hungary, Szego went to
Berlin in 1921. Polya was is Zurich and they started to work on a
problem book. It turned out to contain too much material for one
volume, so was published in two volumes. Polya wrote the following
about their work together on these books.

"It was a wonderful time; we worked with enthusiasm and concentration.
We had similar backgrounds. We were both influenced, like all other
young Hungarian mathematicians of that time, by Leopold Fejer. We were
both readers of the same well directed Hungarian Mathematical Journal
for high school students that stressed problem solving. We were
interested in the same kind of questions, in the same topics; but one
of us knew more about one topic and the other more about some other
topic. It was a fine collaboration. The book `Aufgaben und Lehrsatze
aus der Analysis, I, II', the result of our cooperation, is my best
work and also the best work of Gabor Szego."

It is hard to argue with Polya's assessment about the quality of the
problem books. They set a standard for others who would later write
books of problems and no one has come close to the level achieved by
Polya and Szego. Not only are their problems interesting and
important, they build on each other, so that working the problems in a
section allows the reader to grow and learn new mathematics and
techniques.

Szego spent more than ten years in Germany, first in Berlin as
privatdocent and then in Konigsberg as professor. His first two Ph.D.
students were in Konigsberg. He was a beloved teacher, and when the
situation in Germany became hard and then impossible for Jewish
mathematicians, and Jews in general, Szego was one of the last to
suffer because he was so highly respected by students and colleagues.
While in Berlin, he was awarded the Jules Konig prize in 1924. F.
Riesz gave the report for the prize committee, and this is reprinted
in Riesz's "Oeuvres completes".


In 1934, Gabor Szego moved to the United States, first to St. Louis,
Missouri, where he taught for four years at Washington University, and
then to Stanford, California, where he was chairman of the mathematics
department for 15 years, building it into an excellent department. At
Washington University, Szego wrote the other great book of his, the
first and still the best book on orthogonal polynomials. The study of
these polynomials started in the 19th century, and continues to the
present. In the 1920's, Szego found a variant on the earlier work, and
one of facts he discovered was eventually used in speech synthesisers.

A Ph.D. student from Stanford, Paul Rosenbloom, wrote about life as a
student under Polya and Szego. In addition to the mathematical
education he received, Szego looked out for his cultural education,
giving Rosenbloom a ticket to Bartok's concert at Stanford.

In 1952, Szego published an extension of his first paper. About this
paper Barry McCoy wrote: "It is easily arguable that, of all Szego's
papers, `On Certain Hermitian Forms Associated with the Fourier Series
of a Positive Function' has had the most applications outside of
mathematics. In the first place, the problem which inspired the
theorem was propounded by a chemist working on magnetism. Extensions
of this work made by physicists have led to surprising connections
with integrable systems of nonlinear partial difference and
differential equations.  In addition Szego's theorem has recently been
used by physicists investigating quantum field theory."

One way mathematicians are honored is to have something they
discovered named after them. There is now the Szego kernel function,
the Szego limit theorem and the strong Szego limit theorem, Szego
polynomials orthogonal on the unit circle, the Szego class of
polynomials. Another way we show that the work of a mathematician is
deep enough to last is to publish their selected or collected works.
Szego's "Collected Works" were published in 1982.

I first met Gabor Szego in the 1950's when he returned to St. Louis to
visit old friends, and I was an instructor at Washington University.
Earlier, when I was an undergraduate there, I had used a result found
by Hsu in his Ph.D. thesis at Washington University under Szego. This
was in the first paper I wrote. While at the Univ. of Chicago in the
early 1960's, Szego visited. I still remember seeing him at one end of
the hall and a graduate student, Stephen Vagi, at the other end of the
same hall. They walked toward each other and both started to speak in
Hungarian. I am certain they had not met before, and I have always
wondered how Szego recognized another former Hungarian. In 1972 I
spent a month in Budapest and Szego was there. We talked most days,
and though his health was poor and his memory was not as good as it
had been a few years earlier, we had some very useful discussions.
Three years earlier, also in Budapest, Szego had mentioned two papers
of his which he said should be studied. I did not do it immediately,
but three months later did. One contained the solution of a problem I
had been trying to solve for three years. His paper had been written
40 years earlier. I learned from this that when a great mathematician
tells you to look at a paper of theirs which they think has been
unjustly neglected, one should do it rapidly.


Szego left a memorial for us, his mathematical work. It continues to
live and lead to new work. One of his main areas was orthogonal
polynomials and this is now a very active field. I often regret that
he is not here to appreciate all of the work being done on problems he
started.