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Let (a,b) be any open
sub-interval of the real line, such that −∞≦ a < 0 < b ≦∞. Let Lloc(a,b) be the
space of all the functions which are integrable on each interval (a′,b′) with
a < a′ < b′ < b. There is a one-to-one linear transformation T which maps Lloc(a,b)
into a commutative algebra 𝒜 of (linear) operators. This transformation T maps
convolution into operator-multiplication; therefore, this transformation T is a useful
substitute for the two-sided Laplace transformation; it can be used to solve problems
that are not solvable by the distributional transformations (Fourier or bi-lateral
Laplace).
In essence, the theme of this paper is a commutative algebra 𝒜 of generalized
functions on the interval (a,b); besides containing the function space Lloc(a,b), the
algebra 𝒜 contains every element of the distribution space 𝒟′(a,b) which is regular
on the interval (a,0). The algebra 𝒜 is the direct sum 𝒜−⊕𝒜+, where
𝒜−(respectively, 𝒜+)(a,0) (respectively, to the interval (0,b)). There is a subspace 𝒴
of 𝒜 such that, if y ∈𝒴, then y has an “initial value” ⟨y,0−⟩ and a “derivative” ∂ty
(which corresponds to the usual distributional derivative). If y is a function f() which
is locally absolutely continuous on (a,b), then y belongs to 𝒴, the initial value
⟨y,0−⟩ equals f(O), and ∂ty corresponds to the usual derivative f′(). If y is a
distribution (such as the Dirac distribution) whose support is a locally finite
subset of the interval (a,b), then both y and ∂ty belong to the subspace 𝒴. In
case a = −∞ and b = ∞, the subspace 𝒴 contains the distribution space
𝒟+′.
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