Read e-book online Analytical and Numerical Approaches to Asymptotic Problems PDF

By O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)

ISBN-10: 0080871585

ISBN-13: 9780080871585

ISBN-10: 0444861319

ISBN-13: 9780444861313

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Read or Download Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems PDF

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Extra info for Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems

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For Schwartz functions the result is due to Harish-Chandra and for compactly supported functions to Helgason ( in some special cases) and Gangolli 1 2 1 . In particular the algebra of G-invariant linear operators on S ( € I (K\G/K), * ) is commutative. For example, the G-invariant differential operators D on S are, since they are local operators on K\G, given by convolution with @ E E'(K\G/K) satisfying supp @ C K. But then supp A @ c {O}, that is A @ = Dgt6 for some uniquely determined differential operator D D&is an isomorphism from with constant coefficients on&, and D the algebra Diff (S)G of G-invariant differential operators on S onto the algebra Dif f (a) "61 of W-invariant differential operators with constant coefficients o n m .

In the analytic category, Brjuno [ 2 1 has the statement that the Birkhoff normal form converges on a set of periodic orbits determined by an equation of the form (7). This can be considered as a paraphrase of the Weinstein-Moser method. However, [2] deals also with quasi-periodic solutions, suggesting that the Weinstein-Moser method should have an extension to quasi-periodic solutions as well. Remark 2 . The techniques i) - ii) can also be used to obtain strong asymptotic results (for long time intervals) about all solutions near the equilibrium point (not only the periodic ones).

5 (19721, 143- [ 41 Henrard, J . , Lyapounov's center theorem for resonant equi- librium, J. Diff. Eq. 14 (1973), 431-441. nSral de la stabiliti? Studies 11,Princeton University, 1947. C. van der, On the computation of 4th order coefficients of a normalized Hamiltonian at 1:l resonance, Preprint, Utrecht 1980. [ 71 Moser, J . , Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Corn. Pure Appl. Math. 2 (1976), 727-747. [ 81 Sanders, J . A . , On the theory of nonlinear resonance, Thesis, Utrecht 1978.

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Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems by O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)


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