Approximation of monomials by lower degree polynomials |
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Authors: | D J Newman T J Rivlin |
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Institution: | (1) Department of Mathematics, Yeshiva University, 10033 New York, N.Y., USA;(2) IBM Research Center, P.O. Box 218, 10598 Yorktown Heights, N.Y., USA |
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Abstract: | Our topic is the uniform approximation ofx
k
by polynomials of degreen (n 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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Keywords: | Primary 41A50 41A25 Secondary 41A10 42A08 |
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