APPROXIMATION OF FUNCTIONS BY POLYNOMIALS IN C[-1, 1]

A pointwise estimate for the rate of approximation by polynomials, \f(x) - P(n)(x)\less-than-or-equal-to C(r, lambda)omega(philambda)r(f,n-1delta(n)(x)1-lambda), for 0 less-than-or-equal-to lambda less-than-or-equal-to 1, integer r, phi = square-root 1 - x2 and delta(n)(x) = n-1 + phi(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (lambda = 0) and the recently developed estimate by Totik and first author (lambda = 1). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modem norm estimates for C[-1,1].