View source for Local approximation of functions

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A measure of approximation (in particular, [[Best approximation|best approximation]]) of a function  $  f $
on a set  $  E \subset  \mathbf R  ^ {m} $, 
regarded as a function of this set. The main interest is in the behaviour of a local approximation of a function as  $   \mathop{\rm mes}  E \rightarrow 0 $. 
In certain cases it is possible to characterize the degree of smoothness of the function to be approximated in terms of a local approximation of the function. Let  $  E _ {n} ( f ;  ( \alpha , \beta ) ) $
be the best approximation of a function  $  f \in C [ a , b ] $
by algebraic polynomials of degree  $  n $
on an interval  $  ( \alpha , \beta ) $, 
$  a \leq  \alpha < \beta \leq  b $. 
The following assertion holds: A necessary and sufficient condition for a function  $  f $
to have a continuous derivative of order  $  n + 1 $
at all points of  $  [ a , b ] $
is that

$$ 

\frac{E _ {n} ( f ;  ( \alpha , \beta ) ) }{( \beta - \alpha )  ^ {n+} 1 }

 \rightarrow  \lambda ( x) ,\ \ 
a \leq  x \leq  b ,
$$

uniformly for  $  \beta \rightarrow x $, 
$  \alpha \rightarrow x $, 
$  \alpha < x < \beta $, 
where the continuous function  $  \lambda $
is defined by

$$ 
( n + 1 ) ! 2  ^ {2n+} 1 \lambda ( x)  =  | f ^ { ( n + 1 ) } ( x) | .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.A. Raikov,   "On the local approximation of differentiable functions"  ''Dokl. Akad. Nauk SSSR'' , '''24''' :  7  (1939)  pp. 653–656  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,   "Collected works" , '''2''' , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.A. Brudnyi,   "Spaces defined by means of local approximations"  ''Trans. Moscow Math. Soc.'' , '''24'''  (1974)  pp. 73–139  ''Trudy Moskov. Mat. Obshch.'' , '''24'''  (1971)  pp. 69–132</TD></TR></table>

====Comments====
According to [[#References|[3]]], which is a valuable survey paper with a rather extensive bibliography, the first result characterizing a space of smooth functions in terms of local approximations was obtained by D.A. Raikov [[#References|[1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Peetre,   "On the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060080/l06008018.png" /> spaces"  ''J. Funct. Anal.'' , '''4'''  (1969)  pp. 71–87</TD></TR></table>