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 $\vert f(x)\vert$ and the total variation of $f(x)$ in $[-1,1]$, respectively. Given the function

$$F(x)=F(-1)+\int_{-1}^x f(x)\,dx,$$ (1)

then the coefficients

$$a_n={\textstyle{1\over 2}}(2n+1)\int_{-1}^1 F(x)P_n(x)\,dx$$ (2)

of its Legendre Series, where $P_n(x)$ is a Legendre Polynomial, satisfy the inequalities

$$\vert a_n\vert<\cases{ {6\over\sqrt{\pi}}(M'+V')n^{-3/2} & f... ...q 1$\cr {4\over\sqrt{\pi}}(M'+V')n^{-3/2} & for $n\geq 2$.\cr}$$ (3)

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

Bernstein strengthened Jackson's theorem to

$$2nE_{2n}(\alpha) \leq {4n\over\pi(2n+1)}<{2\over\pi}=0.6366.$$ (4)

A specific application of Jackson's theorem shows that if

$$\alpha(x)=\vert x\vert,$$ (5)

then

$$E_n(\alpha)\leq {6\over n}.$$ (6)

See also Legendre Series, Picone's Theorem

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.