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Jackson's Theorem

Jackson's theorem is a statement about the error E_n(f) of the best uniform approximation to a real function f(x) on [-1,1] by real polynomials of degree at most n. Let f(x) be of bounded variation in [-1,1] and let M^' and V^' denote the least upper bound of |f(x)| and the total variation of f(x) in [-1,1], respectively. Given the function

F(x)=F(-1)+int_(-1)^xf(x)dx,
(1)

then the coefficients

a_n=1/2(2n+1)int_(-1)^1F(x)P_n(x)dx
(2)

of its Fourier-Legendre series, where P_n(x) is a Legendre polynomial, satisfy the inequalities

|a_n|<{6/(sqrt(pi))(M^'+V^')n^(-3/2) for n>=1; 4/(sqrt(pi))(M^'+V^')n^(-3/2) for n>=2.
(3)

Moreover, the Fourier-Legendre series of F(x) converges uniformly and absolutely to F(x) in [-1,1].

Bernstein (1913) strengthened Jackson's theorem to

2nE_(2n)(alpha)<=(4n)/(pi(2n+1))<2/pi=0.6366.
(4)

A specific application of Jackson's theorem shows that if

alpha(x)=|x|,
(5)

then

E_n(alpha)<=6/n.
(6)

See also

Fourier-Legendre Series, Picone's Theorem

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References

Bernstein, S. N. "Sur la meilleure approximation de |x| 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.

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Jackson's Theorem

Cite this as:

Weisstein, Eric W. "Jackson's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacksonsTheorem.html

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