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''of the partial sums of an orthogonal expansion''
 
''of the partial sums of an orthogonal expansion''
  
For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638701.png" />, any orthonormal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638702.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638703.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638704.png" />, the equality
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For any function $  f \in L _ {2} [ a , b ] $,  
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any orthonormal system $  \{ \phi _ {k} \} _ {k=} 1  ^  \infty  $
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on $  [ a , b ] $
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and for any $  n $,  
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the equality
  
<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/m/m063/m063870/m0638705.png" /></td> </tr></table>
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$$
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\inf _
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{\{ a _ {k} \} _ {k=} 1  ^ {n} } \
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\int\limits _ { a } ^ { b }
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\left | f ( x) - \sum _ { k= } 1 ^ { n }  a _ {k} \phi _ {k} ( x) \
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\right |  ^ {2}  d x =
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$$
  
<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/m/m063/m063870/m0638706.png" /></td> </tr></table>
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$$
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= \
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\int\limits _ { a } ^ { b }  | f ( x) - S _ {n} ( f , x ) |  ^ {2}  d x
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$$
  
 
holds, where
 
holds, where
  
<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/m/m063/m063870/m0638707.png" /></td> </tr></table>
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$$
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S _ {n} ( f , x )  = \
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\sum _ { k= } 1 ^ { n }
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c _ {k} ( f  ) \phi _ {k} ( x)
 +
$$
  
is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638708.png" />-th partial sum of the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m0638709.png" /> with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m06387010.png" />, that is,
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is the $  n $-
 +
th partial sum of the expansion of $  f $
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with respect to the system $  \{ \phi _ {k} \} $,  
 +
that is,
  
<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/m/m063/m063870/m06387011.png" /></td> </tr></table>
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$$
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c _ {k} ( f  )  = \
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\int\limits _ { a } ^ { b }
 +
f ( x) \phi _ {k} ( x)  d x .
 +
$$
  
The minimum is attained precisely at the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063870/m06387012.png" /> and
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The minimum is attained precisely at the sum $  S _ {n} ( f , x ) $
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and
  
<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/m/m063/m063870/m06387013.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }  | f ( x) - S _ {n} ( f , x ) |  ^ {2}  d x =
 +
$$
  
<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/m/m063/m063870/m06387014.png" /></td> </tr></table>
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$$
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= \
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\int\limits _ { a } ^ { b }  f ^ { 2 } ( x)  d x - \sum _ { k= } 1 ^ { n }  | c _ {k} ( f  ) |  ^ {2} ,\  n = 1 , 2 ,\dots .
 +
$$
  
 
Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentially are corollaries of this equality (cf. [[Bessel inequality|Bessel inequality]]; [[Parseval equality|Parseval equality]]; [[Complete system of functions|Complete system of functions]]; [[Orthogonal series|Orthogonal series]]; [[Orthogonal system|Orthogonal system]]).
 
Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentially are corollaries of this equality (cf. [[Bessel inequality|Bessel inequality]]; [[Parseval equality|Parseval equality]]; [[Complete system of functions|Complete system of functions]]; [[Orthogonal series|Orthogonal series]]; [[Orthogonal system|Orthogonal system]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1978)  pp. Sect. III.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1978)  pp. Sect. III.4</TD></TR></table>

Revision as of 08:00, 6 June 2020


of the partial sums of an orthogonal expansion

For any function $ f \in L _ {2} [ a , b ] $, any orthonormal system $ \{ \phi _ {k} \} _ {k=} 1 ^ \infty $ on $ [ a , b ] $ and for any $ n $, the equality

$$ \inf _ {\{ a _ {k} \} _ {k=} 1 ^ {n} } \ \int\limits _ { a } ^ { b } \left | f ( x) - \sum _ { k= } 1 ^ { n } a _ {k} \phi _ {k} ( x) \ \right | ^ {2} d x = $$

$$ = \ \int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x $$

holds, where

$$ S _ {n} ( f , x ) = \ \sum _ { k= } 1 ^ { n } c _ {k} ( f ) \phi _ {k} ( x) $$

is the $ n $- th partial sum of the expansion of $ f $ with respect to the system $ \{ \phi _ {k} \} $, that is,

$$ c _ {k} ( f ) = \ \int\limits _ { a } ^ { b } f ( x) \phi _ {k} ( x) d x . $$

The minimum is attained precisely at the sum $ S _ {n} ( f , x ) $ and

$$ \int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x = $$

$$ = \ \int\limits _ { a } ^ { b } f ^ { 2 } ( x) d x - \sum _ { k= } 1 ^ { n } | c _ {k} ( f ) | ^ {2} ,\ n = 1 , 2 ,\dots . $$

Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentially are corollaries of this equality (cf. Bessel inequality; Parseval equality; Complete system of functions; Orthogonal series; Orthogonal system).

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)

Comments

References

[a1] K. Yosida, "Functional analysis" , Springer (1978) pp. Sect. III.4
How to Cite This Entry:
Minimal property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_property&oldid=47842
This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article