## Jackson's Theorem

Jackson'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)

then the coefficients
 (2)

of its Legendre Series, where is a Legendre Polynomial, satisfy the inequalities
 (3)

Moreover, the Legendre Series of converges uniformly and absolutely to in .

Bernstein strengthened Jackson's theorem to

 (4)

A specific application of Jackson's theorem shows that if
 (5)

then
 (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.