TOPICS

# 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)

Fourier-Legendre Series, Picone's Theorem

## Explore with Wolfram|Alpha

More things to try:

## 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.

## Referenced on Wolfram|Alpha

Jackson's Theorem

## Cite this as:

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