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Jan Mikusiński (April 3, 1913 – July 27, 1987) was a Polish mathematician based at the
University of Wrocław The University of Wrocław (, UWr; ) is a public research university in Wrocław, Poland. It is the largest institution of higher learning in the Lower Silesian Voivodeship, with over 100,000 graduates since 1945, including some 1,900 researcher ...
known for his pioneering work in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
. Mikusiński was born in Stanisławów in 1913 and developed an
operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...
, known as the ''Calculus of Mikusiński'' ( MSC 44A40), which is relevant for solving differential equations. His operational calculus is based upon an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
of the convolution of functions with respect to the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
. From the convolution product he goes on to define what in other contexts is called the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fie ...
or a quotient field. These ordered pairs of functions Mikusiński calls "operators", the "Mikusiński operators". He is also well known for Mikusinski's cube, the Antosik–Mikusinski theorem, and Mikusinski convolution algebra. Mikusiński died in
Katowice Katowice (, ) is the capital city of the Silesian Voivodeship in southern Poland and the central city of the Katowice urban area. As of 2021, Katowice has an official population of 286,960, and a resident population estimate of around 315,000. K ...
in 1987. A street in Katowice is named after him.


Selected publications

* ''An Introduction to Analysis - From Number to Integral.'' Wiley 1993 * ''The Operational Calculus.'' Pergamon Press, Oxford 1983 * ''Operatorenrechnung.'' VEB Deutscher Verlag der Wissenschaften, Berlin 1957. * ''The Bochner Integral.'' Birkhäuser 1978. * with Piotr Antosik,
Roman Sikorski Roman Sikorski (July 11, 1920 – September 12, 1983) was a Polish mathematician. Biography Sikorski was a professor at the University of Warsaw from 1952 until 1982. Since 1962, he was a member of the Polish Academy of Sciences. Sikorski's ...
: ''Theory of distributions – the sequential approach.'' Elsevier 1973. * with Stanisław Hartman: ''The theory of Lebesgue Measure and Integration.'' Pergamon Press, Oxford 1961.


See also

* Convolution quotient * Daniell integral


External links


Caricature of Jan Mikusinski

Rachunek Operatorow
(1944)(Operational Calculus ''in Polish'')
Part 1

Part 2
1983 (English)

* {{DEFAULTSORT:Mikusinski, Jan 1913 births 1987 deaths 20th-century Polish mathematicians Recipients of the State Award Badge (Poland) People from Ivano-Frankivsk