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 Fourier-Legendre series, where  is a Legendre polynomial, satisfy the inequalities

(3)

Moreover, the Fourier-Legendre series of  converges uniformly and absolutely to  in .

Bernstein (1913) strengthened Jackson's theorem to

(4)

A specific application of Jackson's theorem shows that if

(5)

then

(6)

REFERENCES:

Bernstein, S. N. "Sur la meilleure approximation de  par les polynomes de degrés donnés." Acta Math. 37, 1-57, 1913.

Cheney, E. W. Introduction to Approximation Theory, 2nd ed. Providence, RI: Amer. Math. Soc., 1999.

Finch, S. R. "Lebesgue Constants." §4.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 250-255, 2003.

Jackson, D. The Theory of Approximation. New York: Amer. Math. Soc., p. 76, 1930.

Korneīĭčuk, N. P. "The Exact Constant in D. Jackson's Theorem on Best Uniform Approximation of Continuous Periodic Functions." Dokl. Akad. Nauk 145, 514-515, 1962.

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.