An algebra of distributions on an open interval

Author:
Harris S. Shultz

Journal:
Trans. Amer. Math. Soc. **169** (1972), 163-181

MSC:
Primary 46F05

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308775-4

MathSciNet review:
0308775

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be any open subinterval of the reals which contains the origin and let denote the family of all distributions on which are regular in some interval , where . Then is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on whose supports are contained in , it is ordinary convolution. Also, 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.

**[1]**J. Horvath,*Topological vector spaces and distributions*. Vol. 1, Addison-Wesley, Reading, Mass., 1966. MR 34 #4863. MR**0205028 (34:4863)****[2]**H. Kestelman,*Modern theories of integration*, 2nd ed., Dover, New York, 1960. MR 23 #A282. MR**0122951 (23:A282)****[3]**G. Krabbe,*Operational calculus*, Springer-Verlag, New York, 1970.**[4]**-,*An algebra of generalized functions on an open interval; two-sided operational calculus*, Bull. Amer. Math. Soc. 77 (1971), 78-84; Correction, ibid., 633. MR 42 #2262; MR 43 #833. MR**0267360 (42:2262)****[5]**-,*Initial-value problems involving generalized functions; two-sided operational calculus*, Arch. Math. (Basel) (to appear).**[6]**-,*A new algebra of distributions; initial-value problems involving Schwartz distributions*(to appear).**[7]**-,*Linear operators and operational calculus*. I, Studia Math. 40 (1971), 199-223.**[8]**H. Shultz,*Linear operators and operational calculus*. II, Studia Math. 41 (to appear).**[9]**F. Treves,*Topological vector spaces, distributions and kernels*, Academic Press, New York, 1967. MR 37 #726. MR**0225131 (37:726)****[10]**G. Krabbe,*An algebra of generalized functions on an open interval; two-sided operational calculus*, Bull. Amer. Math. Soc. 77 (1971), 78-84. MR 42 #2262. MR**0267360 (42:2262)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46F05

Retrieve articles in all journals with MSC: 46F05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308775-4

Keywords:
Generalized functions,
operational calculus,
Schwartz distributions,
two-sided Laplace transformation,
Fourier transformation

Article copyright:
© Copyright 1972
American Mathematical Society