Boehmians Explained
In mathematics, Boehmians are objects obtained by an abstract algebraic construction of "quotients of sequences." The original construction was motivated by regular operators introduced by T. K. Boehme. Regular operators are a subclass of Mikusiński operators, that are defined as equivalence classes of convolution quotients of functions on
. The original construction of Boehmians gives us a space of
generalized functions that includes all regular operators and has the algebraic character of convolution quotients. On the other hand, it includes all
distributions eliminating the restriction of regular operators to
.
Since the Boehmians were introduced in 1981, the framework of Boehmians has been used to define a variety of spaces of generalized functions on
and generalized
integral transforms on those spaces. It was also applied to function spaces on other domains, like
locally compact groups and
manifolds.
The general construction of Boehmians
Let
be an arbitrary nonempty set and let
be a commutative
semigroup acting on
. Let
be a collection of sequences of elements of
such that the following two conditions are satisfied:
(1) If
, then
,
(2) If
and
for some
and all
, then
.
Now we define a set of pairs of sequences:
l{A}=\{((xn),(\phin)):xn\inX,(\phin)\in\Delta,\phimxn=\phinxmforallm,n\inN\}
.
In
we introduce an equivalence relation:
~
if
\phimyn=\psinxmforallm,n\inN
.
The space of Boehmians
is the space of equivalence classes of
, that is
~.
References
- J. Mikusiński, Operational Calculus, Pergamon Press (1959).
- T. K. Boehme, The support of Mikusiński operators, Trans. Amer. Math. Soc. 176 (1973), 319–334.
- J. Mikusiński and P. Mikusiński, Quotients de suites et leurs applications dans l'analyse fonctionnelle (French), [Quotients of sequences and their applications in functional analysis], C. R. Acad. Sci. Paris Sr. I Math. 293 (1981), 463-464.
- P. Mikusiński, Convergence of Boehmians, Japan. J. Math. (N.S.) 9 (1983), 159–179.