- #1

nalkapo

- 28

- 0

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,

**they say on [a,b]=[0,1]**

__at first__f(0)=f(1)=0.

__Why do they want to say that at the end points function must be zero?__

**if the theorem is proved for this case, consider**

__second,__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?