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Approximation of monomials by lower degree polynomials

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Our topic is the uniform approximation ofx k by polynomials of degreen (n<k) on the interval [−1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann 2 and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p k,n , which can be identified with a certain probability. The numberp k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.

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    Riess, R. D. andJohnson, L. W.,Estimates for E n (x n+2m ). Aequationes Math.8 (1972), 258–262.

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Newman, D.J., Rivlin, T.J. Approximation of monomials by lower degree polynomials. Aeq. Math. 14, 451–455 (1976). https://doi.org/10.1007/BF01835995

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AMS (1970) subject classification

  • Primary 41A50, 41A25
  • Secondary 41A10, 42A08