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 ...