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A sequence of algebraic polynomials converging uniformly on  $  [-1, 1] $
to a function  $  f(x) $
that is continuous on this interval. More precisely, Bernstein's interpolation method is a sequence of algebraic polynomials

$$ 
P _ {n} (f;  x)  = \ 

\frac{\sum _ { k=1 } ^ { n }  A _ {k}  ^ {(n)} T _ {n} (x) }{T _ {n} (x _ {k}  ^ {(n)} )(x-x _ {k}  ^ {(n)} ) }
 ,
\  n = 1, 2 \dots
$$

where the

$$ 
T _ {n} (x)  =  \cos ( n   \mathop{\rm arc}  \cos  x)
$$

are the [[Chebyshev polynomials|Chebyshev polynomials]]; the

$$ 
x _ {k}  ^ {(n)}  = \ 
\cos \left [ 
\frac{(2k-1) \pi }{2n}
 \right ]
$$

are the interpolation nodes; and

$$ 
A _ {k}  ^ {(n)}  =  f (x _ {k}  ^ {(n)} )
$$

if  $  k \neq 2ls, l $
is an arbitrary positive integer,  $  n = 2 l q + r $, 
$  q \geq  1 $, 
$  0 \leq  r < 2l $, 
$  s = 1 \dots q; $
otherwise

$$ 
A _ {2ls}  ^ {(n)}  = \ 
\sum _ { i=0 } ^ { l-1 } 
f(x _ {2l (s - 1) + 2i + 1 }   ^ {(n)} ) -
\sum _ { i=1 } ^ { l-1 } 
f (x _ {2l (s - 1) + 2i }   ^ {(n)} ).
$$

The ratio between the degree of the polynomial  $  P _ {n} (f;  x) $
and the number of points at which  $  P _ {n} (f;  x) $
equals  $  f(x) $
is  $  (n - 1)/(n - q) $, 
which tends to  $  2l/(2l - 1) $
as  $  n \rightarrow \infty $; 
if  $  l $
is sufficiently large, this limit is arbitrary close to one. The method was introduced by S.N. Bernstein [S.N. Bernshtein] in 1931 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''2''' , Moscow  (1954)  pp. 130–140  (In Russian)</TD></TR></table>

====Comments====
This method of interpolation seems not very well known in the West. There is, however, a well-known method of Bernstein that uses the special interpolation nodes  $  k/n $, 
$  k = 0 \dots n $, 
for bounded functions on  $  [0, 1] $. 
This method is given by the [[Bernstein polynomials|Bernstein polynomials]]. The sequence of Bernstein polynomials  $  B _ {n} (f;  x) $
constructed for a bounded function  $  f $
on  $  [0, 1] $
converges to  $  f (x) $
at each point of continuity  $  x \in [0, 1] $
of  $  f $. 
If  $  f $
is continuous on  $  [0, 1] $, 
the sequence converges uniformly (to  $  f  $) 
on  $  [0, 1] $. 
If  $  f $
is differentiable,  $  B _ {n} ^ { \prime } (f;  x) \rightarrow f ^ { \prime } (x) $(
at each point of continuity of  $  f ^ { \prime } $), 
cf [[#References|[a1]]].

This method of Bernstein is often used to prove the [[Weierstrass theorem|Weierstrass theorem]] (on approximation). For a generalization of the method (the monotone-operator theorem), see [[#References|[a2]]], Chapt. 3, Sect. 3. See also [[Approximation of functions, linear methods|Approximation of functions, linear methods]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Davis,   "Interpolation and approximation" , Dover, reprint  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>