An algebra of distributions on an open interval
Author:
Harris S. Shultz
Journal:
Trans. Amer. Math. Soc. 169 (1972), 163181
MSC:
Primary 46F05
MathSciNet review:
0308775
Fulltext 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 twosided Laplace transformation.
 [1]
John
Horváth, Topological vector spaces and distributions. Vol.
I, AddisonWesley Publishing Co., Reading, Mass.LondonDon Mills,
Ont., 1966. MR
0205028 (34 #4863)
 [2]
H.
Kestelman, Modern theories of integration, 2nd revised ed.
Dover Publications, Inc., New York, 1960. MR 0122951
(23 #A282)
 [3]
G. Krabbe, Operational calculus, SpringerVerlag, New York, 1970.
 [4]
Gregers
Krabbe, An algebra of generalized functions on
an open interval; twosided operational calculus, Bull. Amer. Math. Soc. 77 (1971), 78–84. MR 0267360
(42 #2262), http://dx.doi.org/10.1090/S000299041971126101
 [5]
, Initialvalue problems involving generalized functions; twosided operational calculus, Arch. Math. (Basel) (to appear).
 [6]
, A new algebra of distributions; initialvalue problems involving Schwartz distributions (to appear).
 [7]
, Linear operators and operational calculus. I, Studia Math. 40 (1971), 199223.
 [8]
H. Shultz, Linear operators and operational calculus. II, Studia Math. 41 (to appear).
 [9]
François
Trèves, Topological vector spaces, distributions and
kernels, Academic Press, New YorkLondon, 1967. MR 0225131
(37 #726)
 [10]
Gregers
Krabbe, An algebra of generalized functions on
an open interval; twosided operational calculus, Bull. Amer. Math. Soc. 77 (1971), 78–84. MR 0267360
(42 #2262), http://dx.doi.org/10.1090/S000299041971126101
 [1]
 J. Horvath, Topological vector spaces and distributions. Vol. 1, AddisonWesley, 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, SpringerVerlag, New York, 1970.
 [4]
 , An algebra of generalized functions on an open interval; twosided operational calculus, Bull. Amer. Math. Soc. 77 (1971), 7884; Correction, ibid., 633. MR 42 #2262; MR 43 #833. MR 0267360 (42:2262)
 [5]
 , Initialvalue problems involving generalized functions; twosided operational calculus, Arch. Math. (Basel) (to appear).
 [6]
 , A new algebra of distributions; initialvalue problems involving Schwartz distributions (to appear).
 [7]
 , Linear operators and operational calculus. I, Studia Math. 40 (1971), 199223.
 [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; twosided operational calculus, Bull. Amer. Math. Soc. 77 (1971), 7884. MR 42 #2262. MR 0267360 (42:2262)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
46F05
Retrieve articles in all journals
with MSC:
46F05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203087754
PII:
S 00029947(1972)03087754
Keywords:
Generalized functions,
operational calculus,
Schwartz distributions,
twosided Laplace transformation,
Fourier transformation
Article copyright:
© Copyright 1972
American Mathematical Society
