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This paper concerns the construction of forms of the error function, en(x) = f(x)- p*n(x), where p*n is the best uniform polynomial approximation of degree n to a continuous function f on [-1, +l]. We show that it is always possible and, from the viewpoint of obtaining explicit results, expedient to write the error as en= a cos(n(Theta + phi), where x =cos Theta, |a|= En(f), the uniform norm of en(x), and the phase angle phi is a continuous function of Theta, depending on f and n. Our classes of explicit best approximations arise from a novel method of determining suitable phase angles in this representation of en(x).

Let T, be the Chebyshev polynomial of the first kind of degree k.