How do you prove the Bolzano-Weierstrass Theorem?
Then snk+1
Are polynomials dense in L2?
Polynomials are dense in weighted L2 space.
What is approximation of a function?
Function approximation is a technique for estimating an unknown underlying function using historical or available observations from the domain. Artificial neural networks learn to approximate a function.
Is the Weierstrass theorem valid for trigonometric polynomials?
The theorem is also valid for real-valued continuous $ 2 \\pi $- periodic functions and trigonometric polynomials, e.g. for real-valued functions which are continuous on a bounded closed domain in an $ m $- dimensional space, or for polynomials in $ m $ variables. For generalizations, see Stone–Weierstrass theorem.
How is the preparation theorem of Weierstrass formulated?
It follows from the Weierstrass preparation theorem that for $ n> 1 $, 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.
Which is the subalgebra of the Weierstrass theorem?
The set of all polynomial functions forms a subalgebra of C [a, b] (that is, a vector subspace of C [a, b] that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C [a, b] .
Which is the lattice version of the Stone Weierstrass theorem?
Stone’s original proof of the theorem used the idea of lattices in C (X, R). A subset L of C (X, R) is called a lattice if for any two elements f, g ∈ L, the functions max { f, g}, min { f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states: Stone–Weierstrass Theorem (lattices).