Summary
We study, in the complex domain, the action of the Fourier transform on the solutions of ordinary linear differential equations with polynomial coefficients. In the classical “Laplace method”, there are some restrictions; also, some choice of integration contours seem rather unsystematic. We show how to remove these restrictions and how to make these choices in a more systematic way.
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.
Litterature
Candelpergher B. — Une introduction à la résurgence,Gazette des Mathématiciens, Soc. Math. Fr., 42 (1989), 36–64.
Ecalle J. — Les fonctions résurgentes,vol. I to III, Publications mathématiques d’Orsay, 1981–85.
Ince E.L.. — Ordinary differential equations, Dover, New-York, 1956.
Kashiwara M. — Systems of microdifferential equations, Progress in math., Birkhaüser, 1983.
Komatsu H. — Operational calculus, hyperfunctions, and ultradistributions, Algebraic analysis (papers dedicated to M. Sato ), Academic Press (1988), 357–372.
Malgrange B. — Introduction aux travaux de J. Ecalle,l’Enseignement mathématique, 31 (1985), 261–282.
Malgrange B. — Equations différentielles linéaires et transformation de Fourier, Ensaios Matemâticos, vol. 1, Soc. Brasil de Matemâtica, 1989.
Malgrange B. — Systèmes holonomes à une variable,(book, to be published).
Malgrange B. — Sur les points singuliers des équations différentielles, l’Enseignement mathématique, 20 fasc.1–2 (1974), 147–176.
Ramis J.-P. — Dévissage Gevrey,Astérisque, 59–60 (1978), 173–204.
Rolba P. — Lemme de Hensel pour des opérateurs différentiels,l’Enseignement mathématique, 26 fasc. 3–4 (1980), 279–311.
Sato M., Kawai T., Kashiwara M. — Hyperfunctions and pseudodifferential equations,Lect. Notes in Math., 287 (1973), 265–529, Springer-Verlag.
Wasow W. — Asymptotic expansions for ordinary differential equations, Interscience publishers, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Malgrange, B. (1991). Fourier Transform and Differential Equations. In: Boutet de Monvel, A., Dita, P., Nenciu, G., Purice, R. (eds) Recent Developments in Quantum Mechanics. Mathematical Physics Studies, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3282-4_2
Download citation
DOI: https://doi.org/10.1007/978-94-011-3282-4_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5449-2
Online ISBN: 978-94-011-3282-4
eBook Packages: Springer Book Archive