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With respect to the approximation of elements in a given set  $  \mathfrak M $, 
the linear method that yields the smallest error among all linear methods. In a normed linear space  $  X $, 
a linear method for the approximation of elements  $  x \in \mathfrak M \subset  X $
by elements of a fixed subspace  $  F \subset  X $
is represented by a linear operator that maps the entire space  $  X $, 
or some linear manifold containing  $  \mathfrak M $, 
into  $  F $. 
If  $  {\mathcal L} $
is the set of all such operators, a best linear method for  $  \mathfrak M $(
if it exists) is defined by an operator  $  \widetilde{A}  \in {\mathcal L} $
for which

$$ 
\sup _ {x \in \mathfrak M } \ 
\| x - \widetilde{A}  x \|  = \ 
\inf _ {A \in {\mathcal L} } \ 
\sup _ {x \in \mathfrak M } \ 
\| x - Ax \| .
$$

The method defined by an operator  $  A $
in  $  {\mathcal L} $
will certainly be a best linear method for  $  \mathfrak M $
relative to the approximating set  $  F $
if, for all  $  x \in \mathfrak M $,

$$ 
\| x - Ax \|  \leq  \ 
\sup _ {x \in \mathfrak M }  E (x, F)
$$

( $  E (x, F) $
is the [[Best approximation|best approximation]] of  $  x $
by  $  F $) 
and if, moreover, for all  $  x \in X $,

$$ 
\| x - Ax \|  =  E (x, F).
$$

The latter is certainly true if  $  X $
is a Hilbert space,  $  F = F _ {n} $
is an  $  n $-
dimensional subspace of  $  X $, 
$  n = 1, 2 \dots $
and  $  A $
is the orthogonal projection onto  $  F _ {n} $, 
i.e.

$$ 
Ax  = \ 
\sum _ {k = 1 } ^ { n }  (x, e _ {k} ) e _ {k} ,
$$

where  $  \{ e _ {1} \dots e _ {n} \} $
is an orthonormal basis in  $  F _ {n} $.

Let  $  X $
be a Banach space of functions defined on the entire real line, with a translation-invariant norm:  $  \| x ( \cdot + \tau ) \| = \| x ( \cdot ) \| $(
this condition holds, e.g. for the norms of the spaces  $  C = C [0, 2 \pi ] $
and  $  L _ {p} = L _ {p} (0, 2 \pi ) $, 
$  1 \leq  p \leq  \infty $, 
of  $  2 \pi $-
periodic functions); let  $  F = T _ {n} $
be the subspace of trigonometric polynomials of order  $  n $. 
There exist best linear methods (relative to  $  T _ {n} $) 
for a class  $  \mathfrak M $
of functions  $  x (t) \in X $
that contains  $  x (t + \alpha ) $
for any  $  \alpha \in \mathbf R $
whenever it contains  $  x (t) $. 
An example is the linear method

$$ \tag{* }
A (x;  t;  \mu , \nu )  =  
\frac{\mu _ {0} a _ {0} }{2}
 +
$$

$$ 
+ 
\sum _ {k = 1 } ^ { n }  \{ \mu _ {k} (a _ {k}  \cos  kt + b _ {k} \ 
\sin  kt) + \nu _ {k} (a _ {k}  \sin  kt - b _ {k}  \cos  kt) \} ,
$$

where  $  a _ {k} $
and  $  b _ {k} $
are the Fourier coefficients of  $  x (t) $
relative to the trigonometric system, and  $  \mu _ {k} $
and  $  \nu _ {k} $
are numbers.

Now consider the classes  $  W _  \infty   ^ {r} M $(
and  $  W _ {1}  ^ {r} M $), 
$  r = 1, 2 \dots $
of  $  2 \pi $-
periodic functions  $  x (t) $
whose derivatives  $  x ^ {(r - 1) } (t) $
are locally absolutely continuous and whose derivatives  $  x  ^ {(r)} (t) $
are bounded in norm in  $  L _  \infty  $(
respectively, in  $  L _ {1} $) 
by a number  $  M $. 
For these classes, best linear methods of the type (*) yield the same error (over the entire class) in the metric of  $  C $(
respectively,  $  L _ {1} $) 
as the best approximation by a subspace  $  T _ {n} $; 
the analogous assertion is true for these classes with any rational number  $  r > 0 $(
interpreting the derivatives  $  x  ^ {(r)} (t) $
in the sense of Weyl). For integers  $  r = 1, 2 \dots $
best linear methods of type (*) have been constructed using only the coefficients  $  \mu _ {k} $(
all  $  \nu _ {k} = 0 $).

If  $  F = S _ {n}  ^ {r} $
is the subspace of  $  2 \pi $-
periodic polynomial splines of order  $  r $
and defect 1 with respect to the partition  $  k \pi /n $, 
$  k = 0, \pm  1 \dots $
then a best linear method for the classes  $  W _  \infty  ^ {r + 1 } M $(
and  $  W _ {1} ^ {r + 1 } $), 
$  r = 1, 2 \dots $
is achieved in  $  L _ {p} $, 
$  1 \leq  p \leq  \infty $(
resp. in  $  L _ {1} $) 
by splines in  $  S _ {n}  ^ {r} $
interpolating the function  $  x (t) $
at the points  $  k \pi /n + [1 + (-1)  ^ {r} ] \pi /4n $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Korneichuk,   "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.M. Tikhomirov,   "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Kiesewetter,   "Vorlesungen über lineare Approximation" , Deutsch. Verlag Wissenschaft.  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Rice,   "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley  (1964)</TD></TR></table>