Let
(
a
,
b
)
(a,b)
be any open subinterval of the reals which contains the origin and let
B
\mathfrak {B}
denote the family of all distributions on
(
a
,
b
)
(a,b)
which are regular in some interval
(
∈
,
0
)
( \in ,0)
, where
∈>
0
\in > 0
. Then
B
\mathfrak {B}
is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on
(
a
,
b
)
(a,b)
whose supports are contained in
[
0
,
b
)
[0,b)
, it is ordinary convolution. Also,
B
\mathfrak {B}
can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.