Weiertrass Approximation Theorem
Weiertrass Approximation Theorem : Statement : If f is a real continuous function defined on a closed interval [a,b] then there exists a sequence of real polynomials {Pₙ} which converges uniformly to f(x) on [a,b] i.e lim Pₙ(x)=f(x) n-->∞ converges uniformly on [a,b] . Proof : If a=b , the conclusion follows by taking Pₙ(x) to be a constant polynomial , defined by Pₙ(x) = f(a) for all n . We may thus assume that a<b . We next observe that a linear transformation x' = (x-a) / (b-a) is a continuous mapping of [a,b] onto [1,0] . Accordingly , we assume without loss of generality that a=0 , b= 1 . Consider F(x) = f(x) - f(0) - x[f(1) - f(0)] , for 0≤x≤1 Hence F(0) = 0= F(1) , and if F can be expressed as a limit of uniformly convergent sequence of polynomials , then the same is true for f , since f-F is a pol