# What are they doing here? (polynomials-functions)

## Main Question or Discussion Point

I am currently studying Stone–Weierstrass approximation theorem, and some parts really confuse me.
basically, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.

In the proof,
at first they say on [a,b]=[0,1]
f(0)=f(1)=0. Why do they want to say that at the end points function must be zero?

second, if the theorem is proved for this case, consider
g(x)= f(x)- f(0) - x[f(1)-f(0)] (0<= x <= 1)
Here g(0)=g(1)=0

Can anyone explain the meaning and purpose of these transformation?

My guess is that they want to define function f and then find a g(x) which looks like polynomials. Is that right? and also, is there any relationship between these transformation and uniformly convergence?

HallsofIvy
Homework Helper
I am currently studying Stone–Weierstrass approximation theorem, and some parts really confuse me.
basically, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.

In the proof,
at first they say on [a,b]=[0,1]
f(0)=f(1)=0. Why do they want to say that at the end points function must be zero?
I don't understand what you are saying. What is "f"? What are the hypotheses here?

second, if the theorem is proved for this case, consider
g(x)= f(x)- f(0) - x[f(1)-f(0)] (0<= x <= 1)
Here g(0)=g(1)=0

Can anyone explain the meaning and purpose of these transformation?

My guess is that they want to define function f and then find a g(x) which looks like polynomials. Is that right? and also, is there any relationship between these transformation and uniformly convergence?

I don't understand what you are saying. What is "f"? What are the hypotheses here?
f is any continuous function f(x) on an interval [a,b] . in the proof they chose [0,1] .
they want to prove that for any continuous function, there is a polynomial that can be approximated to this function as much as desired.
i couldn't understand the change between those function.

mathman