Being a promising young mathematician, he was sent on a two-year
specialization in Germany, mainly in Leipzig and Göttingen
(1910-1912), where at that time had been working D. Hilbert, F.
Klein, E. Landau, H. Weyl. Göttingen was the mathematical world
center. There, being influenced by Landau, Tschakaloff became
absorbed by Number Theory, and Complex Analysis. His scientific
accomplishments from that period had been presented in his
habilitation work ``Analytical Characteristics of the Riemann
Function z(z)''.
In 1922 Tschakaloff was elected a full-time professor in mathematics
at Sofia University. A second postgraduate specialization in the
period 1924-1925 gave him the opportunity to work in Paris, Pisa,
and Naples. At the University of Naples he wrote and defended his
PhD dissertation on ``Riccati Equations''. Tricomi described it as
``a work of significant value''. After his return to Bulgaria in
1925 he was elected a correspondent member of the Bulgarian Academy
of Sciences, and in 1930 he became a regular member of the Academy.
Since 1922 till his death in 1963 he chaired the Advanced Analysis
Department of Sofia University. He was Dean of the Faculty of
Mathematics and Physics in 1923-1924 and Rector of Sofia University
in 1943-1944. He was the first Director of the Mathematical
Institute.
Lubomir Tschakaloff was member of the Physics and Mathematics Union.
He was member of the Royal Czech Academy of Sciences, the Geography
Union in Lima, the Warsaw Academy of Sciences.
He died on September 11, 1963.
Tschakaloff's basic contribution to science is in the field of
Function Theory, Number Theory, and Quadrature Formulae.
His studies on ``squarable lunes'' expanded Landau's results and
directed another great mathematician's attention - that of
Tschebotareff, to the complete solution of Klausen's problem from
1840.
Tschakaloff studied the integral representation and applications of
Newton's quotients (divided differences), founded and described
basic principles and recurrent relations of their Peano kernels,
which much later the famous American mathematician Isaac Schoenberg
called B-splines. They became the fundament of the emerging in the
'60s theory of spline functions. This was one of the most
significant developments in applied analysis during that century.
The first results in this field Tschakaloff announced on the
International Mathematical Congress in Oslo in 1936. Unfortunately,
being ahead of their time, his publications passed unnoticed for a
long time. Furthermore, Tschakaloff was the first to explain the
integral representation of B-splines.
In a series of publications, Lubomir Tschakaloff studied Gaussian
quadrature formulae with multiple nodes, offered original
methods for the calculation of their coefficients and representation
of the remainders, and proved the existence of the Gaussian nodes.
These studies had led to difficult extremal problems that were in
the focus of attention of specialists in classical and applied
analysis for decades. Typical example is the problem of uniqueness
of the optimal Gaussian nodes that was completely solved in 1975.
Tschakaloff found out the exact limits of the subinterval, where the
indeterminate point in the Roll's theorem lies, when the
function considered was a polynomial of any fixed degree. Thus he
solved the known Pompeiu problem. This result brought him a
world-wide fame. The method suggested by Tschakaloff and developed
in later times by Favar remained in literature as Tschakaloff-Favar
approach.
Tschakaloff was the author of 110 scientific publications. He wrote
books on theory of analytical functions and differential equations.
Among his students and followers have been Acad. Nikola Obrechkoff,
Acad. Lubomir Iliev, Prof. Yaroslav Tagamlitski, Prof. Arkady
Stoyanov.
