Chebyshev theorem

2020 Mathematics Subject Classification: Primary: 41A50 [MSN][ZBL]

If a function $f(x)$ is continuous on $[a,b]$ and if

$$A=\max_{a\leq x\leq b}|f(x)-P_n(x)|,$$

$$P_n(x)=\sum_{k=0}^na_kx^k,$$

then $P_n(x)$ is the polynomial of best uniform approximation for $f(x)$, i.e.

$$\max_{a\leq x\leq b}|f(x)-P_n(x)|=\min_{\{c_k\}}\max_{a\leq x\leq b}\left|f(x)-\sum_{k=0}^nc_kx^k\right|,$$

if and only if there exist $n+2$ points $a\leq x_0<\dots<x_{n+1}\leq b$ in Chebyshev alternation, which means that the condition

$$f(x_i)-P_n(x_i)=\epsilon A(-1)^i,\quad i=0,\dots,n+1,$$

is satisfied, where $\epsilon=1$ or $-1$. This theorem was proved by P.L. Chebyshev in 1854 (cf. [1]) in a more general form, namely for the best uniform approximation of functions by rational functions with fixed degrees of the numerator and denominator. Chebyshev's theorem remains valid if instead of algebraic polynomials one considers polynomials

$$P_n(x)=\sum_{k=0}^nc_k\phi_k(x),$$

where $\{\phi_k(x)\}_{k=0}^n$ is a Chebyshev system. The criterion formulated in Chebyshev's theorem leads to methods for the approximate construction of polynomials of best uniform (Chebyshev) approximation. In a somewhat different formulation Chebyshev's theorem can be extended to functions of a complex variable (cf. [2]) and to abstract functions (cf. [3]).

References

[1]	P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 151–238 (In Russian)
[2]	A.N. Kolmogorov, "A remark on the polynomials of Chebyshev deviating the least from a given function" Uspekhi Mat. Nauk , 3 : 1 (1948) pp. 216–221 (In Russian)
[3]	S.I. Zukhovitskii, S.B. Stechkin, "On the approximation of abstract functions with values in a Banach space" Dokl. Akad. Nauk SSSR , 106 : 5 (1956) pp. 773–776 (In Russian)
[4]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)

References

[a1]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2