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With respect to the approximation of elements in a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159301.png" />, the linear method that yields the smallest error among all linear methods. In a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159302.png" />, a linear method for the approximation of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159303.png" /> by elements of a fixed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159304.png" /> is represented by a linear operator that maps the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159305.png" />, or some linear manifold containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159306.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159307.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159308.png" /> is the set of all such operators, a best linear method for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b0159309.png" /> (if it exists) is defined by an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593010.png" /> for which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593011.png" /></td> </tr></table>
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The method defined by an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593013.png" /> will certainly be a best linear method for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593014.png" /> relative to the approximating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593015.png" /> if, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593016.png" />,
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593017.png" /></td> </tr></table>
+
$$
 +
\sup _ {x \in \mathfrak M } \
 +
\| x - \widetilde{A}  x \|  = \
 +
\inf _ {A \in {\mathcal L} } \
 +
\sup _ {x \in \mathfrak M } \
 +
\| x - Ax \| .
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593018.png" /> is the [[Best approximation|best approximation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593019.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593020.png" />) and if, moreover, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593021.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593022.png" /></td> </tr></table>
+
$$
 +
\| x - Ax \|  \leq  \
 +
\sup _ {x \in \mathfrak M }  E (x, F)
 +
$$
  
The latter is certainly true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593023.png" /> is a Hilbert space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593024.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593025.png" />-dimensional subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593028.png" /> is the orthogonal projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593029.png" />, i.e.
+
( $  E (x, F) $
 +
is the [[Best approximation|best approximation]] of  $  x $
 +
by  $  F $)
 +
and if, moreover, for all  $  x \in X $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593030.png" /></td> </tr></table>
+
$$
 +
\| x - Ax \|  = E (x, F).
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593031.png" /> is an orthonormal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593032.png" />.
+
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.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593033.png" /> be a Banach space of functions defined on the entire real line, with a translation-invariant norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593034.png" /> (this condition holds, e.g. for the norms of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593037.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593038.png" />-periodic functions); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593039.png" /> be the subspace of trigonometric polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593040.png" />. There exist best linear methods (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593041.png" />) for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593042.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593043.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593044.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593045.png" /> whenever it contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593046.png" />. An example is the linear method
+
$$
 +
Ax  = \
 +
\sum _ {k = 1 } ^ { n }  (x, e _ {k} ) e _ {k} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
where  $  \{ e _ {1} \dots e _ {n} \} $
 +
is an orthonormal basis in  $  F _ {n} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593048.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593050.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593051.png" /> relative to the trigonometric system, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593053.png" /> are numbers.
+
$$ \tag{* }
 +
A (x;  t;  \mu , \nu )  =
 +
\frac{\mu _ {0} a _ {0} }{2}
 +
+
 +
$$
  
Now consider the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593054.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593055.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593057.png" />-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593058.png" /> whose derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593059.png" /> are locally absolutely continuous and whose derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593060.png" /> are bounded in norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593061.png" /> (respectively, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593062.png" />) by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593063.png" />. For these classes, best linear methods of the type (*) yield the same error (over the entire class) in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593064.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593065.png" />) as the best approximation by a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593066.png" />; the analogous assertion is true for these classes with any rational number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593067.png" /> (interpreting the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593068.png" /> in the sense of Weyl). For integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593069.png" /> best linear methods of type (*) have been constructed using only the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593070.png" /> (all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593071.png" />).
+
$$
 +
+
 +
\sum _ {k = 1 } ^ { n }  \{ \mu _ {k} (a _ {k}  \cos  kt + b _ {k} \
 +
\sin  kt) + \nu _ {k} (a _ {k}  \sin  kt - b _ {k}  \cos  kt) \} ,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593072.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593073.png" />-periodic polynomial splines of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593074.png" /> and defect 1 with respect to the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593076.png" /> then a best linear method for the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593077.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593078.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593079.png" /> is achieved in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593081.png" /> (resp. in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593082.png" />) by splines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593083.png" /> interpolating the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593084.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015930/b01593085.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 10:58, 29 May 2020


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 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

[1] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[2] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[3] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

Comments

References

[a1] H. Kiesewetter, "Vorlesungen über lineare Approximation" , Deutsch. Verlag Wissenschaft. (1973)
[a2] J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
How to Cite This Entry:
Best linear method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_linear_method&oldid=19004
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article