Abstract
Various additive formulae for supports and singular supports of convolutions are studied in terms of the Fourier transform in complex domain. Part of the material presented below was announced in our note [4]. The authors with to thank Professor L. Hörmander for his most hepful comments.
The first author was partially supported by the U.S.Army Office of Research (Durham).
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
ATIYAH, M.F. & MACDONALD, I.G., "Introduction to Commutative Algebra", Addison-Wesley, Reading, Mass., 1969.
BERENSTEIN, C.A. — Convolution operators and related quasianalytic classes, Ph. D. Thesis, New York University, 1970.
BERENSTEIN, C.A. & DOSTAL, M.A. — "Analytically Uniform Spaces and their Applications to Convolution Equations", Lecture Notes in Mathematics, vol. 256, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
BERENSTEIN, C.A. & DOSTAL, M.A. — Sur une classe de fonctions entières, C.R. Acad. Sc. Paris, série A, 274 (1972), p. 1149–1152.
BERENSTEIN, C.A. & DOSTAL, M.A. — Some remarks on convolution equations, to appear in Ann. Inst. Fourier.
BOCHNER, S. — "Vorlesungen über Fouriersche Integrale", Leipzig, 1932.
DE WILDE, M. — Une propriété de relèvement et espaces à réseaux absorbants, C.R. Acad. Sc. Paris, série A, 266 (1968), p. 457–459.
DOSTAL, M.A. — On a property of convolution, Comm. Pure Appl. Math., 20 (1967), p. 595–607.
DOSTAL, M.A. — An analogue of a theorem of Vladimir Berenstein and its applications to singular supports of distributions, Proc. London Math. Soc., 19 (1969), p. 553–576.
DOSTAL, M.A. — A complex characterization of the Schwartz space (μ), Math. Ann., 195 (1972), p. 175–191.
DOSTAL, M.A. — Some recent results in the theory of topological vector spaces, to appear in "Proceedings of the Symposium on Analysis, Recife (1972), Springer Verlag, 1973.
EHRENPREIS, L. — Solutions of some problems of division IV, Amer. J. Math., 82 (1960), p. 522–588.
EHRENPREIS, L. — "Fourier Analysis in Several Complex Variables", Wiley-Interscience, New York, 1970.
HERZ, C.S. — Fourier transforms related to convex sets, Ann. of Math., 75 (1962) p. 81–92.
HORMANDER, L. — On the range of convolution operators, Ann. of Math., 76 (1962) p. 148–170.
HORMANDER, L. — Hypoelliptic convolution equations, Math. Scand., 9 (1961), p. 178–184.
HORMANDER, L. — Supports and singular supports of convolutions, Acta Math., 110 (1965), p. 279–302.
LIONS, J.L. — Supports dans la transformation de Laplace, J. d'Analyse Math., 2 (1952–1953), p. 369–380.
MALGRANGE, B. — Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955–1956), p. 271–355.
MIKUSIŃSKI, J. — "Operational Calculus", Pergamon Press, New York, 1959.
RONKIN, L.I. — "Introduction into the Theory of Entire Functions of Several Variables" (in Russian), Moscow, 1971.
SCHWARTZ, L. — "Théorie des distributions", I, II, Hermann, Paris, 1966.
WATSON, G.N. — "A Treatrise on the Theory of Bessel Functions", 2nd edition, Cambridge University Press, 1966.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1973 Springer-Verlag
About this paper
Cite this paper
Berenstein, C.A., Dostal, M.A. (1973). On convolution equations I. In: Akutowicz, E.J. (eds) L'Analyse Harmonique dans le Domaine Complexe. Lecture Notes in Mathematics, vol 336. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065789
Download citation
DOI: https://doi.org/10.1007/BFb0065789
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06392-6
Online ISBN: 978-3-540-47170-7
eBook Packages: Springer Book Archive