National Institute of the Republic of Serbia

Slobodan Aljančić, professor at the Faculty of Sciences and Mathematics in Belgrade for many years, and a full member of the Serbian Academy of Sciences and Arts from the age of 46, was one of the most outstanding mathematicians and teachers in Serbia in the second half of the 20th century, and also a highly respected and loved man.

He was born in Belgrade, on March 12, 1922. His father, Zdenko Aljančić, was of Slovenian origin, and his mother Bisenija came from a Belgrade family. He finished primary and secondary school in Belgrade. In 1940, he started his studies of civil engineering. After World War II he continued his education at the Faculty of Philosophy, the Mathematics Department from which he graduated in 1947.

Aljančić started research work at the Faculty already, and
published his first paper in 1948. After graduation he became a secondary
school teacher, and also a part-time assistant at the Faculty of Natural
Sciences until 1951, when he obtained a permanent position of an assistant at
the Faculty. Continuing his research, in 1953 he obtained his Ph.D. title
from the Academy of Sciences defending his thesis *On Asymptotic Expansions
of A-Summable Linear Functionals*. His further scientific development and
his university career were rather quick. He became assistant professor in
1954, associate professor in 1959 and full professor in 1968. He was elected
a corresponding member of the Serbian Academy of Sciences and Arts in 1961,
and a full member in 1968.

Almost from the very beginning of his scientific career,
Aljančić was part of the circle of students and associates of the
great Serbian mathematician Jovan Karamata, working in particular with
Miodrag Tomić and Ranko Bojanić. He studied, alone or with the
group, primarily summability of trigonometric series and regularly varying
functions, and before that he worked on asymptotic series and approximation
theory. While dealing with them he closely worked with the leading expert in
this area, Jacques Favard, thanks to whom he was on sabbatical in Paris
during the 1957/58 school year. The friendship with Favard continued in the
subsequent years. Apart from his stay in Paris, Aljančić also spent
three months in the United States, in 1971 and attended several domestic and
foreign conferences and congresses. His teaching career was also remarscable
and he, among other things, published two successful text-books. One of them,
the *Introduction to Real and Functional Analysis* was used by numerous
generations of students.

During the last two decades of his life, Aljančić suffered from a serious heart condition but due to discipline and his perseverance he successfully coped with it. Regrettably, he did not recover from a malignant disease which struck him later on. He died on March 19, 1993.

The work of Aljančić can be classified into the following five areas:

I. The theory of asymptotic series, II. Approximation theory, III. Trigonometric series, IV. Summability, V. Regularly varying functions.

I Asymptotic series was introduced by Poincare and Stieltjes at the end of the 19th century, and later on this notion was generalized in several ways. The first papers of Aljančić partially belong to this area, and the dissertation mentioned above was entirely dedicated to problems from this area. In other articles, Aljančić obtained several new asymptotic expansions.

II The approximation theory deals with estimation of the elements of a space of real functions by certain classes of trigonometric polynomials and the results consist mostly of direct and inverse theorems, equivalence theorems, and saturation theorems. Aljančić started working on this theory in 1957 and, as already mentioned, he worked with Favard. In his works dealing with this theory, he studied various approximation procedures, especially in connection with saturation classes and orders of approximation for saturations.

III In the area of trigonometric series Aljančić, alone or together with Bojanić and Tomić., used slowly varying functions to generalize, among others, theorems concerned with the asymptotic behavior of sine and cosine series and with the relationship between the integrability of some functions and convergence of the corresponding series. He also studied certain problems in connection with Fourier series, transformed by multiplication of their given coefficients.

IV The study of summability has a long and abundant tradition in Serbia.
Before and after World War II, Karamata and many of his followers conducted
through research of these problems, which resulted in the so-called direct (*Abelian*),
inverse (*Tauberian*) and Mercerian theorems. Aljančić
obtained important results dealing with summability, mainly in relation to
the Mercerian theorem.

V Aljančić's works that dealt with regular variation (regularly varying functions) are among his most important works. Slowly (and regularly) varying functions were introduced by Karamata in the early 1930s. Karamata set the foundation of this theory and determined its various applications. Later on, several authors, from Serbia and abroad, further developed this theory and its various applications. Together with Bojanić and Tomić, with Karamata, or alone, Aljančić applied these functions to the investigation of the behavior of sums of trigonometric series, to the problem of integrability of these sums and to a problem concerning Frullani integral. Two of his articles deal with the theory of regular variation in a broad sense. The first one, worked on together with Bojanić and Tomić, contains a systematic study of the so-called slowly varying functions with a remainder term, while the second one, which he worked on with D. Arandjelović, proposes the theory of the so-called O-regularly varying functions. These functions were considered by Karamata and Avakumović already in the 1930s, but Aljančić and Arandjelović went much further in their investigations and also found some applications.

There is much evidence in existence which proves that many years of teaching of Slobodan Aljancic were as important as his scientific achievements. The quality of his lectures and standard of his teaching deserve a special mention. Besides, he wrote several very good textbooks, but only had two published. He was mentor to several Ph.D. and masters candidates, who provided his students with indispensable help. He is also responsible, and respected as well, for introducing functional analysis in Serbia.

As a man and colleague, Prof. Aljančić possessed special kindness and warmth and always behaved calmly and considerately towards others.

Slobodan Aljančić, the academician and professor, was a central link, or the so-called bridge between the generation of Serbian mathematicians of the past on the one hand; whose leading figures were Mihailo Petrović, Bogdan Gavrilović, Nikolaj Saltikov, Tadija Pejović, Jovan Karamata, Miloš Radojčić, Radivoje Kašanin, Vojislav Avakumović and Dragoljub Marković and the generation of mathematicians of the present era on the other.

His bibliography is found here.

Authors: Adamović, Dušan; Aranđelović, Dragan