Simple Proof of Weierstass Approximation Theorem?

In summary, we are given a set D of real numbers and a function g(x) defined on D. We need to find a function G(x) that is continuous everywhere and equal to g(x) when x is in set D. To do this, we can use the definition of continuity and L'Hospital's rule to find the limits of g(x) at certain points. We can also refer to Riemann's theorem on removable singularities for guidance. The Weirstrass Approximation Thm. may also be useful in this case.
  • #1
toddlinsley79
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0

Homework Statement


Let D={x in the set of real numbers: -3<x<3, x does not equal 0,1,2} and define g(x)=(cosx-1)/x + (x3-2x2-x+2)/(x2-3x+2) on D. Find G:R→R such that G is continuous everywhere and G(x)=g(x) when x is in set D.


Homework Equations





The Attempt at a Solution



From a past homework problem I know how to prove that, for any continuous f:R→R, there exists a sequence (pn) of polynomials such that pn converges uniformly to f on any given bounded subset of R.
So after I show that g(x) is continuous and that a sequence of polynomials that converges uniformly to g exists, how do I actually find the function G(x)?
 
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  • #2
Use the definition of continuity: g(0) = lim(x->0) g(x) = 1 etc. (The limits of the form 0/0 are to be tackled with L'Hospital's rule).
You might want to have a look at Riemann's theorem on removable singularities to see how this is done in general.
P.S. - Where is the Weirstrass Approximation Thm. called for?
 

What is the Weierstrass Approximation Theorem?

The Weierstrass Approximation Theorem states that any continuous function on a closed interval can be approximated by a polynomial function.

Why is the Weierstrass Approximation Theorem important?

The Weierstrass Approximation Theorem is important because it allows us to approximate any continuous function with a polynomial function, which is often easier to work with in mathematical calculations.

How is the Weierstrass Approximation Theorem proved?

The Weierstrass Approximation Theorem is typically proved using the Stone-Weierstrass Theorem, which states that any continuous function on a compact space can be approximated by a polynomial function.

What are the applications of the Weierstrass Approximation Theorem?

The Weierstrass Approximation Theorem has applications in various fields such as physics, engineering, and economics. It is also used in numerical analysis to approximate functions and solve differential equations.

Are there any limitations to the Weierstrass Approximation Theorem?

Yes, the Weierstrass Approximation Theorem only guarantees the existence of an approximation, but not the uniqueness or efficiency of the approximation. Additionally, the degree of the polynomial required for a good approximation may be very large for some functions, making it computationally challenging.

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