View source for Chebyshev approximation

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''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 [[#References|[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|best approximation]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 23–51  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Guter,   L.D. Kudryavtsev,   B.M. Levitan,   "Elements of the theory of functions" , Moscow  (1963)  (In Russian)</TD></TR></table>



====Comments====
See also [[#References|[a1]]], especially Chapt. 3, and [[#References|[a2]]], Section 7.6. For an obvious reason, Chebyshev approximation is also called best uniform approximation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,   "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126</TD></TR></table>