Polynomial Approximations to Continuous Functions

where g(x) is an increasing continuous function such that g(0) = 0 and g ( 1 n+2 ) > an. If p is a polynomial such that ||p − f ||∞ < an, then for each of the points xi = i where i ∈ {1, 2, . . . , n + 2}, we have |p(xi) − f (xi)| < an and f (xi) = (−1)ig ( 1 i ) . Since g ( 1 i ) ≥ g ( 1 n+2 ) > an, this means that f (xi) and p(xi) have the same sign. Thus, the sign of p(xi) alternates with each i, and by the Intermediate Value Theorem p, has a root in the interval (xi, xi+1). This makes a total of n + 1 roots, so the degree of p is greater than n, which means that en(f ) > an. There is an equivalent construction in [1]. There, the author uses a function defined as an infinite sum of Chebyshev polynomials.