Vol. 42, No. 2, 1972

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An algebra of generalized functions on an open interval: two-sided operational calculus

Gregers Louis Krabbe

Vol. 42 (1972), No. 2, 381–411

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+.

Mathematical Subject Classification 2000
Primary: 46F10
Secondary: 46E30, 46H05
Received: 19 August 1971
Published: 1 August 1972
Gregers Louis Krabbe