TIMAN'S TYPE RESULT ON APPROXIMATION BY ALGEBRAIC POLYNOMIALS
YURI BRUDNYI

The paper presents two approximation results of the Jackson-Timan
type, which were included in my lecture at the All-Union
Conference in Approximation Theory (Dniepropetrovsk, June
26-28, 1990). The conference was dedicated to the seventieth
birthday of Professor Alexander Timan (26.6.1920-13.8.1988).
Unfortunately, proceedings of the conference never appeared
because of the well-known events in the former Soviet Union, and
only the abstracts were published (see [#3]).  Here we present
the approximation method which has been used in the proof of
these results.  In fact, it can be applied to many other
approximation problems of such a kind.

In view of the character of this volume, let me give some details
regarding Timan's life, which has many common features with
Glazman's life.  Professor Timan belonged to the same unfortunate
generation as Professor Glazman, with whom he was closely
acquainted.  More precisely, they were included in the small part
of that generation which was lucky enough to survive.

Professor Alexander Timan was born in the small Ukrainian town
Zolotonosha.  At the age of sixteen, Alexander Timan entered the
Dnepropetrovsk State University.  He started his creative work
under the strong influence of academicians Kolmogorov and Kagan,
who had positions at the university.  After graduating in 1941,
Alexander Timan passed the entrance examinations for postdoctoral
studies at the Moscow State University.  The Second World War
drastically changed his plans.  From September 1941 until the end
of the war, Alexander Timan was an artillery officer at the front
line.  As Glazman, he took part in many heavy battles, including
the Moscow military operation of 1941 and the capture of Berlin
in May 1945.

After returning to Dnepropetrovsk he started working at the
university.  In 1952, he became a professor and head of the
Chair of Function Theory and Geometry. His mathematical
achievements during that fruitful period of his scientific
career were summarized in his classical book "Approximation
Theory of Real Functions" (the third edition was published
recently, see [T'2]).

In 1961, high ranking party officials put pressure on the
university administration to dismiss him from his university
position.  The official reason given was some dispute with
another colleague.  But the real reason was his Jewish origins
with which he strongly identified.  This event had an extremely
negative effect on his mathematical creativity and on the
subsequent scientific careers of his Jewish students.  (One of
them lost his job almost immediately, after trying to defend his
teacher.)

For the rest of his life, Alexander Timan worked as a professor
at the Dnepropetrovsk Institute of Chemical Technology.