Weierstrass' infinite product theorem [1]: For any given sequence of points in the complex plane ,

(1)

there exists an entire function with zeros at the points of this sequence and only at these points. This function may be constructed as a canonical product:

(2)

where is the multiplicity of zero in the sequence (1), and The multipliers are called Weierstrass prime multipliers or elementary factors. The exponents are chosen so as to ensure the convergence of the product (2); for instance, the choice ensures the convergence of (2) for any sequence of the form (1).

It also follows from this theorem that any entire function with zeros (1) has the form where is the canonical product (2) and is an entire function (see also Hadamard theorem on entire functions).

Weierstrass' infinite product theorem can be generalized to the case of an arbitrary domain : Whatever a sequence of points without limit points in , there exists a holomorphic function in with zeros at the points and only at these points.

The part of the theorem concerning the existence of an entire function with arbitrarily specified zeros may be generalized to functions of several complex variables as follows: Let each point of the complex space , , be brought into correspondence with one of its neighbourhoods and with a function which is holomorphic in . Moreover, suppose this is done in such a way that if the intersection of the neighbourhoods of the points is non-empty, then the fraction is a holomorphic function in . Under these conditions there exists an entire function in such that the fraction is a holomorphic function at every point . This theorem is known as Cousin's second theorem (see also Cousin problems).

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Weierstrass' theorem on the approximation of functions: For any continuous real-valued function on the interval there exists a sequence of algebraic polynomials which converges uniformly on to the function ; established by K. Weierstrass .

Similar results are valid for all spaces . The Jackson theorem is a strengthening of this theorem.

The theorem is also valid for real-valued continuous -periodic functions and trigonometric polynomials, e.g. for real-valued functions which are continuous on a bounded closed domain in an -dimensional space, or for polynomials in variables. For generalizations, see Stone–Weierstrass theorem. For the approximation of functions of a complex variable by polynomials, see [3].

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Weierstrass' theorem on uniformly convergent series of analytic functions : If the terms of a series

(*)

which converges uniformly on compacta inside a domain of the complex plane , are analytic functions in , then the sum is an analytic function in . Moreover, the series obtained by successive term-by-term differentiations of the series (*), for any , also converges uniformly on compacta inside towards the derivative of the sum of the series (*). This theorem has been generalized to series of analytic functions of several complex variables converging uniformly on compacta inside a domain of the complex space , , and the series of partial derivatives of a fixed order of the terms of the series (*) converges uniformly to the respective partial derivative of the sum of the series:

Weierstrass' theorem on uniform convergence on the boundary of a domain : If the terms of a series are continuous in a closed bounded domain of the complex plane and are analytic in , then uniform convergence of this series on the boundary of the domain implies that it converges uniformly on the closed domain .

This property of series of analytic functions is also applicable to analytic and harmonic functions defined, respectively, in a domain of the complex space , , or in the Euclidean space , . As a general rule it remains valid in all situations in which the maximum-modulus principle is applicable.

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Weierstrass' preparation theorem. A theorem obtained and originally formulated by K. Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation whose left-hand side is a holomorphic function of two complex variables. This theorem generalizes the following important property of holomorphic functions of one complex variable to functions of several complex variables: If is a holomorphic function of in a neighbourhood of the coordinate origin with , , then it may be represented in the form , where is the multiplicity of vanishing of at the coordinate origin, , while the holomorphic function is non-zero in a certain neighbourhood of the origin.

The formulation of the Weierstrass preparation theorem for functions of complex variables, . Let be a holomorphic function of in the polydisc and let Then, in some polydisc the function can be represented in the form where is the multiplicity of vanishing of the function at the coordinate origin, ; the functions are holomorphic in the polydisc the function is holomorphic and does not vanish in . The functions , , and are uniquely determined by the conditions of the theorem.

If the formulation is suitably modified, the coordinate origin may be replaced by any point of the complex space . It follows from the Weierstrass preparation theorem that for , as distinct from the case of one complex variable, every neighbourhood of a zero of a holomorphic function contains an infinite set of other zeros of this function.

Weierstrass' preparation theorem is purely algebraic, and may be formulated for formal power series. Let be the ring of formal power series in the variables with coefficients in the field of complex numbers ; let be a series of this ring whose terms have lowest possible degree , and assume that a term of the form , , exists. The series can then be represented as where are series in whose constant terms are zero, and is a series in with non-zero constant term. The formal power series and are uniquely determined by .

A meaning which is sometimes given to the theorem is the following division theorem: Let the series satisfy the conditions just specified, and let be an arbitrary series in . Then there exists a series and series which satisfy the following equation:

Weierstrass' preparation theorem also applies to rings of formally bounded series. It provides a method of inductive transition, e.g. from to . It is possible to establish certain properties of the rings and in this way, such as being Noetherian and having the unique factorization property. There exists a generalization of this theorem to differentiable functions [6].

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The polynomial which occurs in the Weierstrass preparation theorem, is called a Weierstrass polynomial of degree in .

The analogue of the Weierstrass preparation theorem for differentiable functions is variously known as the differentiable preparation theorem, the Malgrange preparation theorem or the Malgrange–Mather preparation theorem. Let be a smooth real-valued function on some neighbourhood of in and let with and smooth near in . Then the Malgrange preparation theorem says that there exists a smooth function near zero such that for suitable smooth , and the Mather division theorem says that for any smooth near in there exist smooth functions and on near such that with . For more sophisticated versions of the differentiable preparation and division theorems, cf. [a2]–[a4].

An important application is the differentiable symmetric function theorem (differentiable Newton theorem), which says that a germ of a symmetric differentiable function of in can be written as a germ of a differentiable function in the elementary symmetric functions , , [a7], [a8].

There exist also -adic analogues of the preparation and division theorems. Let be a complete non-Archimedean normed field (cf. Norm on a field). is the algebra of power series , , , , such that as , . The norm on is defined by . The subring consists of all with and is the ideal of all with . Let be the residue ring , and let be the quotient mapping. Then , where is the residue field of . An element with is called regular in of degree if is of the form with and . Note that is naturally a subalgebra of . The -adic Weierstrass preparation and division theorem now says: i) (division) Let be regular of degree in and let . Then there exist unique elements and , , such that and, moreover, , where ; ii) (preparation) Let be of norm , then there exists a -automorphism of such that is regular in .

References

[a1]	L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4
[a2]	M. Golubitsky,   "Stable mappings and their singularities" , Springer  (1973)  pp. Chapt. IV
[a3]	J.C. Tougeron,   "Ideaux de fonction différentiables" , Springer  (1972)  pp. Chapt. IX
[a4]	B. Malgrange,   "Ideals of differentiable functions" , Oxford Univ. Press  (1966)  pp. Chapt. V
[a5]	J. Fresnel,   M. van der Put,   "Géométrie analytique rigide et applications" , Birkhäuser  (1981)  pp. §II.2
[a6]	N. Koblitz,   "-adic numbers, -adic analysis, and zeta-functions" , Springer  (1977)  pp. 97
[a7]	G. Glaeser,   "Fonctions composés différentiables"  Ann. of Math. , 77  (1963)  pp. 193–209
[a8]	S. Łojasiewicz,   "Whitney fields and the Malgrange–Mather preparation theorem"  C.T.C. Wall (ed.) , Proc. Liverpool Singularities Symposium I , Lect. notes in math. , 192 , Springer  (1971)  pp. 106–115