I am currently studying Stone–Weierstrass approximation theorem, and some parts really confuse me.(adsbygoogle = window.adsbygoogle || []).push({});

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 firstthey 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?

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

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