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Intrinsic nilpotent approximation. (English) Zbl 0621.22010

From the author’s introduction: ”This report is a preliminary version of work to date on an approximation process arising in the context of constructing an appropriate non-isotropic ”perturbation” theory for certain classes of naturally arising linear differential operators P. This requires the construction of approximate state spaces or phase spaces. These will depend on the structure of P itself, and may vary locally, i.e. from point to point of the base manifold M, or microlocally, i.e. from point to point of the cotangent space. A basic issue is that the structure of P itself determines the minimal amount of information that the initial approximation must contain, and this may vary from point to point.”
The paper has the nice feature that it is informal, and written in a leisurely style. The introduction is long and informative; it acquaints the reader with a large slice of the modern history of partial differential equations.
On the one hand, the paper formulates and proves no theorems. On the other hand, it has a lot of interesting calculations and examples. It suggests some nice open problems.
Reviewer: S.G.Krantz

MSC:

22E25 Nilpotent and solvable Lie groups
17B30 Solvable, nilpotent (super)algebras
35B20 Perturbations in context of PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
17B65 Infinite-dimensional Lie (super)algebras
65H10 Numerical computation of solutions to systems of equations
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