Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.
Before addressing this question, think of the case of polynomials. In one dimension, Weierstrass theorem for polynomials says that there exists a polynomial with prescribed zeros. In several dimensions, zeros are never isolated. So what does it mean "prescribed zeros"? Zeros of a polynomial form an "algebraic set". But an algebraic set is defined as... zero set of a polynomial. We obtain a tautology.
However there is something which can be called a "generalization of Weierstrass theorem". This is called the "solution of the Second Cousin Problem". See any book on several complex variables for exact formulation.
For example, the Weierstrass theorem has the following consequence: every meromorphic function in the whole plane is a ratio of two entire functions. (Here a meromorphic function is defined as a function which is LOCALLY (in a neighborhod of every point) can be represented as a ratio of two analytic functions in a neighborhood of this point. So the statement is not a tautology.)
The same statement is true in any dimension and it is proved using the solution of the second Cousin problem. In this sense the second Cousin problem is the generalization of the Weierstrass theorem to several variables.