Abstract
As exemplified in the introduction of L. Schwartz’ book [24] individual generalized functions had been known before he introduced the distributions D′ (Ω) in 1948–51. Dirac’s δ function and Hadamard’s finite part of x α+ , α < -1, are famous. More generally P. Lévy [16] had determined the dual of C m([a, b]). Yet the true history of generalized functions started with Schwartz.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Bengel and P. Schapira, Décomposition microlocale analytique des distributions, Ann. Inst Fourier, Grenoble 29–3 (1979), 101–124.
G. Björck, Linear partial differential operators and generalized distributions, Art f. Mat. 6 (1966), 351–407.
J.-M. Bony, Propagation des singularités différentiables pour une classe d’opérateurs différentiels à coefficients analytiques, Astérisque 34–35 (1976), 43–91.
J.-M. Bony and P. Schapira, Solutions hyperfunctions du problème de Cauchy, Lecture Notes in Math. 287 (1973), 82–98.
J. W. De Roever, Hyperfunctional singular support of ultradistributions, J. Fac. Sci. Univ. Tokyo, Sect IA 31 (1984), 585–631.
A. Eida, On microlocal decomposition of ultradistributions and ultradifferentiable functions, Master’s thesis, Univ. Tokyo (1989).
A. Eida, Harmonic analysis of ultradistributions, Doctor’s thesis, Univ. Tokyo, (1992).
A. Grothendieck, Sur les espaces de solutions d’une classe générale d’équations aux dérivées partielles, J. Analyse Math. 2 (1952–53), 243–280.
L. Hörmander, “An Introduction to Complex Analysis in Several Variables”, Van Nostrand, Princeton (1966).
L. Hörmander, “The Analysis of Linear Partial Differential Operators I”, Springer, Berlin-Heidelberg-New York-Tokyo (1983).
A. Kaneko, “Introduction to Hyperfunctions”, Kluwer Academic, Dordrecht-Boston-London (1988).
H. Komatsu, On the index of ordinary differential operators, J. Fac. Sci. Univ. Tokyo, Sect IA 18 (1971), 379–398.
H. Komatsu, Ultradistributions I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. IA 20 (1973), 25–105.
H. Komatsu, Microlocal Analysis in Gevrey classes and in complex domains, in: “Microlocal Analysis and Applications”, Lecture Notes in Math. 1495 (1991), 161–236.
H. Komatsu, An elementary theory of hyperfunctions and microfunctions, Banach Center Publ. 27 (1992), to appear.
P. Lévy, “Leçon d’Analyse Fonctionnel”, Gauthier-Villard, Paris (1922).
A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, in: “Theory of Distributions”, Inst. Gulbenkian de Ciência, Lisboa (1964), 195–326.
A. Martineau, Le “edge of the wedge theorem” en théorie des hyperfonctions de Sato, in: “Proc. Internat. Conf. on Functional Analysis and Related Topics (1969)”, Univ. of Tokyo Press (1970), 95–106.
C. Roumieu, Sur quelques extensions de la notion de distribution, Ann. Ecole Norm. Sup. 77 (1960), 41–121.
M. Sato, Theory of hyperfunctions, I, J. Fac. Sci. Univ. Tokyo, Sect. I 8 (1959), 139–193.
M. Sato, Theory of hyperfunctions, II, J. Fac. Sci. Univ. Tokyo, Sect. I 8 (1960), 387–437.
M. Sato, Hyperfunctions and partial differential equations, in: “Proc. Internat. Conf. on Functional Analysis and Related Topics, 1969”, Univ. of Tokyo Press (1970), 91–94.
M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Lecture Notes in Math. 287 (1973), 265–529.
L. Schwartz, “Théorie des Distributions“, Hermann, Paris (1950–51).
J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math. 29 (1974), 67–73.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Komatsu, H. (1993). Hyperfunctions, Ultradistributions and Microfunctions. In: Pathak, R.S. (eds) Generalized Functions and Their Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1591-7_11
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1591-7_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1593-1
Online ISBN: 978-1-4899-1591-7
eBook Packages: Springer Book Archive