Composite quasianalytic functions
| Autor: | Michael Chow, Edward Bierstone, André Belotto da Silva |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Composite function
Pure mathematics Algebra and Number Theory Mathematics - Complex Variables 03C64 26E10 32S45 (Primary) 30D60 32B20 (Secondary) 010102 general mathematics Composite number Neighbourhood (graph theory) Resolution of singularities Mathematics - Logic Continuation theorem Function (mathematics) Composition (combinatorics) 01 natural sciences Mathematics - Classical Analysis and ODEs 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics Point (geometry) 010307 mathematical physics Complex Variables (math.CV) 0101 mathematics Logic (math.LO) Mathematics |
| Zdroj: | Compositio Mathematica. 154:1960-1973 |
| ISSN: | 1570-5846 0010-437X |
| Popis: | We prove two main results on Denjoy-Carleman classes: (1) a composite function theorem which asserts that a function f(x) in a quasianalytic Denjoy-Carleman class Q, which is formally composite with a generically submersive mapping y=h(x) of class Q, at a single given point in the source (or in the target) of h, can be written locally as f(x) = g(h(x)), where g(y) belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar loss of regularity for functions definable in the o-minimal structure given by expansion of the real field by restricted functions of quasianalytic class Q. Both results depend on an estimate for the regularity of an infinitely differentiable solution g of the equation f(x) = g(h(x)), with f and h as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood. Comment: 13 pages |
| Databáze: | OpenAIRE |
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