Weierstrass preparation theorem
In the following we use the standard notation for coordinates in that . That is is the first coordinates.
Let be a function analytic in a neighbourhood of the origin such that extends to be analytic at the origin and is not zero at the origin for some positive integer (in other words, as a function of , the function has a zero of order at the origin). Then there exists a polydisc such that every function holomorphic and bounded in can be written as
Note that is not necessarily a Weierstrass polynomial.
Let be as above, then there is a unique representation of as , where is analytic in a neighbourhood of the origin and and being a Weierstrass polynomial.
It should be noted that the condition that extends to be analytic, which is equivalent to saying that , is not an essential restriction. In fact , then there exists a linear change of coordinates, arbitrarily close to the identity, such that the condition of the theorem is satisfied in the new set of coordinates.
- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Title||Weierstrass preparation theorem|
|Date of creation||2013-03-22 15:04:28|
|Last modified on||2013-03-22 15:04:28|
|Last modified by||jirka (4157)|
|Synonym||Weierstrass division theorem|