# Difference between revisions of "Chebyshev approximation"

uniform approximation

Approximation of a continuous function $f$ defined on a set $M$ by functions $S$ from a given class of functions, where the measure of approximation is the deviation in the uniform metric

$$\rho(f,S)=\sup_{x\in M}|f(x)-S(x)|.$$

P.L. Chebyshev in 1853 [1] raised and studied the problem of best uniform approximation of a continuous function by algebraic polynomials of degree not exceeding $n$. For this problem, and also for the more general problem concerning best uniform approximation of a function by rational functions, he obtained fundamental results, and at the same time laid the foundations for the theory of best approximation.

#### References

 [1] P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 23–51 (In Russian) [2] R.S. Guter, L.D. Kudryavtsev, B.M. Levitan, "Elements of the theory of functions" , Moscow (1963) (In Russian)