| 释义 |
Jackson's TheoremJackson's theorem is a statement about the error  of the best uniform approximation to a Real Function  on  by Real Polynomials of degree at most  . Let be of bounded variation in and let  and  denote the least upper bound of  and the total variation of in , respectively. Given the function
 | (1) |
 | (2) |
 is a Legendre Polynomial, satisfy the inequalities
 | (3) |
 converges uniformly and absolutely to in .
Bernstein strengthened Jackson's theorem to
 | (4) |
 | (5) |
 | (6) |
References
Cheney, E. W. Introduction to Approximation Theory. New York: McGraw-Hill, 1966.Jackson, D. The Theory of Approximation. New York: Amer. Math. Soc., p. 76, 1930. Rivlin, T. J. An Introduction to the Approximation of Functions. New York: Dover, 1981. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 205-208, 1991.
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