Weierstrassteoremet

The Weierstrassteoremet refers to several fundamental results in real and complex analysis attributed to Karl Weierstrass. The most prominent among them are the Weierstrass approximation theorem, the Weierstrass product (factorization) theorem, and the Weierstrass preparation theorem; the Stone–Weierstrass theorem is often cited as a broader generalization related to the first.

Weierstrass approximation theorem states that every continuous function on a closed interval can be uniformly approximated as closely as desired by polynomials. In its classical form, it implies that polynomials are dense in the space of continuous functions on [a,b]. A broader generalization is provided by the Stone–Weierstrass theorem: if X is a compact Hausdorff space and A is a subalgebra of C(X) that contains the constants and separates points, then A is dense in C(X).

The Weierstrass product (factorization) theorem concerns entire functions in complex analysis. It asserts that given a prescribed sequence of zeros (with possible multiplicities) that has no finite accumulation point, there exists an entire function with exactly those zeros. Such functions can be represented as products of canonical factors, known as Weierstrass products, possibly multiplied by an exponential factor. The theorem provides a canonical way to encode the zeros of an entire function into a product expansion.

The Weierstrass preparation theorem is a local factorization result in several complex variables (and one variable). It states that near a zero of a holomorphic function, the function can be written as a product of a holomorphic polynomial (the Weierstrass polynomial) and a unit (a nonvanishing holomorphic function). This result underpins division and factorization in the local ring of holomorphic functions and is key to analytic continuation and the structure of analytic sets.

Together, these results form a central part of Weierstrass’s influence on analysis, providing tools for approximation, factorization, and local behavior of holomorphic functions.