View source for Best complete approximation

<!--
b0159201.png
$#A+1 = 38 n = 0
$#C+1 = 38 : ~/encyclopedia/old_files/data/B015/B.0105920 Best complete approximation
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A best approximation of a function  $  f (x _ {1} \dots x _ {k} ) $
in  $  k $
variables  $  (k \geq  2) $
by algebraic or trigonometric polynomials. Let  $  X $
be the space  $  C $
or  $  L _ {p} $
of functions  $  f (x _ {1} \dots x _ {k} ) $, 
$  2 \pi $-
periodic in each variable, that are either continuous or  $  p $-
summable ( $  p \geq  1 $) 
on the  $  k $-
dimensional period cube with edges of length  $  2 \pi $.

The best complete approximation of a function  $  f (x _ {1} \dots x _ {k} ) \in X $
by trigonometric polynomials is the quantity

$$ 
E _ {n _ {1}  \dots n _ {k} }
(f) _ {X}  = \ 
\inf _ {T _ {n _ {1}  \dots n _ {k} } } \ 
\| f - T _ {n _ {1}  \dots n _ {k} } \| _ {X} ,
$$

where the infimum is taken over all trigonometric polynomials of degree  $  n _ {i} $
in the variable  $  x _ {i} $(
$  1 \leq  i \leq  k $). 
Together with the best complete approximation, one also considers best partial approximations.

A best partial approximation of a function  $  f (x _ {1} \dots x _ {k} ) \in X $
is a best approximation by functions  $  T _ {n _ {\nu _ {1}  } \dots n _ {\nu _ {r}  } } (x _ {1} \dots x _ {k} ) \in K $
that are trigonometric polynomials of degree  $  n _ {\nu _ {1}  } \dots n _ {\nu _ {r}  } $(
$  1 \leq  r < k $), 
respectively, in the fixed variables  $  x _ {\nu _ {1}  } \dots x _ {\nu _ {r}  } $
with coefficients depending on the remaining  $  k - r $
variables, i.e.

$$ 
E _ {n _ {\nu _ {1}  } \dots n _ {\nu _ {r}  } , \infty } (f) _ {X}  = \ 
\inf _
{T _ {n _ {\nu _ {1}  } \dots n _ {\nu _ {r}  } } } \ 
\| f - T _ {n _ {\nu _ {1}  } \dots n _ {\nu _ {r}  } } \| _ {X} .
$$

It is obvious that

$$ 
E _ {n _ {1}  \dots n _ {r} \dots n _ {k} } (f) _ {X}  \geq  \ 
E _ {n _ {1}  \dots n _ {r} , \infty } (f) _ {X} .
$$

S.N. Bernstein [S.N. Bernshtein] [[#References|[1]]] proved the following inequality for continuous functions in two variables:

$$ \tag{1 }
E _ {n _ {1}  , n _ {2} } (f) _ {C}  \leq  \ 
$$

$$ 
\leq   A   \mathop{\rm ln} (2 +  \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) (E _ {n _ {1}  , \infty } (f) _ {C} + E _ {n _ {2}  , \infty } (f) _ {C} ),
$$

where  $  A $
is an absolute constant. It has been shown [[#References|[3]]] that the term  $   \mathop{\rm ln} (2 +  \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) $
in inequality (1) (and in the analogous relation for the space  $  L _ {1} $) 
cannot be replaced by a factor with a slower rate of increase as  $   \mathop{\rm min} \{ n _ {1} , n _ {2} \} \rightarrow \infty $.

In the space  $  L _ {p} $(
$  p > 1 $) 
one has the inequality

$$ \tag{2 }
E _ {n _ {1}  \dots n _ {k} } (f) _ {L _ {p}  }  \leq  \ 
A _ {p, k } 
\sum _ {i = 1 } ^ { k }  E _ {n _ {i}  , \infty } (f) _ {L _ {p}  } ,
$$

where the constant  $  A _ {p, k }  $
depends only on  $  p $
and  $  k $.

Similar definitions yield best complete approximations and best partial approximations of functions defined on a closed bounded domain  $  \Omega \subset  \mathbf R  ^ {k} $
by algebraic polynomials, and in this case inequalities similar to (1) and (2) have been established.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,   "Collected works" , '''2''' , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.N. Temlyakov,   "On best approximations of functions in two variables"  ''Dokl. Akad. Nauk SSSR'' , '''223''' :  5  (1975)  pp. 1079–1082  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>