Preparation theorems for matrix valued functions

Nils Dencker

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 3, page 865-892
  • ISSN: 0373-0956

Abstract

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We generalize the Malgrange preparation theorem to matrix valued functions F ( t , x ) ∈ C ∞ ( R × R n ) satisfying the condition that t ↦ det F ( t , 0 ) vanishes to finite order at t = 0 . Then we can factor F ( t , x ) = C ( t , x ) P ( t , x ) near (0,0), where C ( t , x ) ∈ C ∞ is inversible and P ( t , x ) is polynomial function of t depending C ∞ on x . The preparation is (essentially) unique, up to functions vanishing to infinite order at x = 0 , if we impose some additional conditions on P ( t , x ) . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation and division theorems.

How to cite

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Dencker, Nils. "Preparation theorems for matrix valued functions." Annales de l'institut Fourier 43.3 (1993): 865-892. <http://eudml.org/doc/75023>.

@article{Dencker1993,
abstract = {We generalize the Malgrange preparation theorem to matrix valued functions $F(t,x)\in C^\infty (\{\bf R\}\times \{\bf R\}^n)$ satisfying the condition that $t\mapsto \, \{\rm det\}\, F(t,0)$ vanishes to finite order at $t=0$. Then we can factor $F(t,x) = C(t,x)P(t,x)$ near (0,0), where $C(t,x)\in C^\infty $ is inversible and $P(t,x)$ is polynomial function of $t$ depending $C^\infty $ on $x$. The preparation is (essentially) unique, up to functions vanishing to infinite order at $x=0$, if we impose some additional conditions on $P(t,x)$. We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation and division theorems.},
author = {Dencker, Nils},
journal = {Annales de l'institut Fourier},
keywords = {normal forms; elementary divisors; Malgrange preparation theorem; matrix valued functions},
language = {eng},
number = {3},
pages = {865-892},
publisher = {Association des Annales de l'Institut Fourier},
title = {Preparation theorems for matrix valued functions},
url = {http://eudml.org/doc/75023},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Dencker, Nils
TI - Preparation theorems for matrix valued functions
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 3
SP - 865
EP - 892
AB - We generalize the Malgrange preparation theorem to matrix valued functions $F(t,x)\in C^\infty ({\bf R}\times {\bf R}^n)$ satisfying the condition that $t\mapsto \, {\rm det}\, F(t,0)$ vanishes to finite order at $t=0$. Then we can factor $F(t,x) = C(t,x)P(t,x)$ near (0,0), where $C(t,x)\in C^\infty $ is inversible and $P(t,x)$ is polynomial function of $t$ depending $C^\infty $ on $x$. The preparation is (essentially) unique, up to functions vanishing to infinite order at $x=0$, if we impose some additional conditions on $P(t,x)$. We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation and division theorems.
LA - eng
KW - normal forms; elementary divisors; Malgrange preparation theorem; matrix valued functions
UR - http://eudml.org/doc/75023
ER -

References

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  1. [1] N. DENCKER, The Propagation of Polarization in Double Refraction, J. Funct. Anal., 104 (1992), 414-468. Zbl0767.58041MR93e:35108
  2. [2] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators I-IV, Springer-Verlag, Berlin, 1983-1985. Zbl0612.35001
  3. [3] B. MALGRANGE, Le théorème de préparation en géométrie différentiable, Séminaire H. Cartan, 15, 1962-1963, Exposés 11, 12, 13, 22. Zbl0119.28501
  4. [4] B. MALGRANGE, The preparation theorem for differentiable functions, Differential Analysis, 203-208, Oxford University Press, London, 1964. Zbl0137.03601MR32 #178
  5. [5] B. MALGRANGE, Ideals of differentiable functions, Oxford University Press, London, 1966. Zbl0177.17902
  6. [6] J. MATHER, Stability of C∞ mappings : I. The division theorem, Ann. of Math., 87 (1968), 89-104. Zbl0159.24902MR38 #726
  7. [7] J. MATHER, Stability of C∞ mappings, III : Finitely determined map-germs, Publ. Math. I.H.E.S., 35 (1968), 127-156. Zbl0159.25001MR43 #1215a
  8. [8] L. NIRENBERG, A proof of the Malgrange preparation theorem, Springer Lecture Notes in Math., 192 (1971), 97-105. Zbl0212.10702MR54 #586
  9. [9] B. L. van der WAERDEN, Algebra II, 5 Aufl., Springer-Verlag, Berlin, 1967. Zbl0192.33002MR38 #1968

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