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An equality expressing the square of the norm of an element in a vector space with a scalar product in terms of the square of the moduli of the [[Fourier coefficients|Fourier coefficients]] of this element in some [[Orthogonal system|orthogonal system]]. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715901.png" /> is a normed separable vector space with a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715902.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715903.png" /> is the corresponding norm and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715904.png" /> is an orthogonal system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715907.png" /> then Parseval's equality for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p0715908.png" /> is
+
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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/p/p071/p071590/p0715909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159011.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159012.png" /> in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159014.png" /> is orthonormal, then Parseval's equality has the form
+
An equality expressing the square of the norm of an element in a vector space with a scalar product in terms of the square of the moduli of the [[Fourier coefficients|Fourier coefficients]] of this element in some [[Orthogonal system|orthogonal system]]. Thus, if  $  X $
 +
is a normed separable vector space with a scalar product  $  (  , ) $,
 +
if  $  \| \cdot \| $
 +
is the corresponding norm and if  $  \{ e _ {n} \} $
 +
is an orthogonal system in  $  X $,
 +
$  e _ {n} \neq 0 $,
 +
$  n = 1, 2 \dots $
 +
then Parseval's equality for an element  $  x \in X $
 +
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/p/p071/p071590/p07159015.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\| x \|  ^ {2}  = \sum _ { n= } 1 ^  \infty  | a _ {n} |  ^ {2} \| e _ {n} \|  ^ {2} ,
 +
$$
  
The validity of Parseval's equality for a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159016.png" /> is a necessary and sufficient condition for its Fourier series in the orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159017.png" /> to converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159018.png" /> in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159019.png" />. The validity of Parseval's equality for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159020.png" /> is a necessary and sufficient condition for the orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159021.png" /> to be complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159022.png" /> (cf. [[Complete system|Complete system]]). This implies, in particular, that:
+
where  $  a _ {n} = ( x, e _ {n} )/( e _ {n} , e _ {n} ) $,
 +
$  n = 1, 2 \dots $
 +
are the Fourier coefficients of  $  x $
 +
in the system $  \{ e _ {n} \} $.  
 +
If  $  \{ e _ {n} \} $
 +
is orthonormal, then Parseval's equality has the form
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159023.png" /> is a separable Hilbert space (cf. [[Hilbert space|Hilbert space]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159024.png" /> is an orthogonal basis of it, then Parseval's equality holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159025.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159026.png" />;
+
$$
 +
\| x \|  ^ {2}  = \sum _ { n= } 1 ^  \infty  | a _ {n} | ^ {2} .
 +
$$
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159027.png" /> is a separable Hilbert space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159028.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159029.png" /> is an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159030.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159032.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159034.png" />, then
+
The validity of Parseval's equality for a given element  $  x \in X $
 +
is a necessary and sufficient condition for its Fourier series in the orthogonal system  $  \{ e _ {n} \} $
 +
to converge to  $  x $
 +
in the norm of $  X $.  
 +
The validity of Parseval's equality for every element  $  x \in X $
 +
is a necessary and sufficient condition for the orthogonal system  $  \{ e _ {n} \} $
 +
to be complete in  $  X $(
 +
cf. [[Complete system|Complete system]]). This implies, in particular, that:
  
<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/p/p071/p071590/p07159035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
1) if  $  X $
 +
is a separable Hilbert space (cf. [[Hilbert space|Hilbert space]]) and  $  \{ e _ {n} \} $
 +
is an orthogonal basis of it, then Parseval's equality holds for  $  \{ e _ {n} \} $
 +
for every  $  x \in X $;
 +
 
 +
2) if  $  X $
 +
is a separable Hilbert space,  $  x , y \in X $,
 +
if  $  \{ e _ {n} \} $
 +
is an orthonormal basis of  $  X $
 +
and if  $  a _ {n} = ( x, e _ {n} ) $
 +
and  $  b _ {n} = ( y, e _ {n} ) $
 +
are the Fourier coefficients of  $  x $
 +
and  $  y $,
 +
then
 +
 
 +
$$ \tag{2 }
 +
( x, y)  = \sum _ { n= } 1 ^  \infty  a _ {n} \overline{ {b _ {n} }}\; ,
 +
$$
  
 
the so-called generalized Parseval equality. In a fairly-definitive form the question of the completeness of a system of functions that are the eigen functions of differential operators was studied by V.A. Steklov in [[#References|[1]]].
 
the so-called generalized Parseval equality. In a fairly-definitive form the question of the completeness of a system of functions that are the eigen functions of differential operators was studied by V.A. Steklov in [[#References|[1]]].
  
Parseval's equality can also be generalized to the case of non-separable Hilbert spaces: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159038.png" /> is a certain index set), is a complete orthonormal system in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159039.png" />, then for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159040.png" /> Parseval's equality holds:
+
Parseval's equality can also be generalized to the case of non-separable Hilbert spaces: If $  \{ e _  \alpha  \} $,  
 +
$  \alpha \in \mathfrak A $(
 +
$  \mathfrak A $
 +
is a certain index set), is a complete orthonormal system in a Hilbert space $  X $,  
 +
then for any element $  x \in X $
 +
Parseval's equality holds:
  
<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/p/p071/p071590/p07159041.png" /></td> </tr></table>
+
$$
 +
( x, x)  = \sum _ {\alpha \in \mathfrak A } | ( x, e _  \alpha  ) |  ^ {2} ,
 +
$$
  
 
and the sum on the right-hand side is to be understood as
 
and the sum on the right-hand side is to be understood as
  
<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/p/p071/p071590/p07159042.png" /></td> </tr></table>
+
$$
 +
\sup _ {\mathfrak A _ {0} }  \sum _ {\alpha \in \mathfrak A } | ( x, e _  \alpha  ) |  ^ {2} ,
 +
$$
  
where the supremum is taken over all finite subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159044.png" />.
+
where the supremum is taken over all finite subsets $  \mathfrak A _ {0} $
 +
of $  \mathfrak A $.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159045.png" />, the space of real-valued functions with Lebesgue-integrable squares on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159046.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159047.png" />, then one may take the [[Trigonometric system|trigonometric system]] as a complete orthogonal system and
+
When $  X = L _ {2} [- \pi , \pi ] $,  
 +
the space of real-valued functions with Lebesgue-integrable squares on $  [- \pi , \pi ] $,  
 +
and $  f \in L _ {2} [- \pi , \pi ] $,  
 +
then one may take the [[Trigonometric system|trigonometric system]] as a complete orthogonal system 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/p/p071/p071590/p07159048.png" /></td> </tr></table>
+
$$
 +
f  \sim 
 +
\frac{a _ {0} }{2}
 +
+ \sum _ { n= } 1 ^  \infty 
 +
( a _ {n}  \cos  nx + b _ {n}  \sin  nx),
 +
$$
  
 
where (1) takes the form
 
where (1) takes the form
  
<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/p/p071/p071590/p07159049.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1} \pi
 +
\int\limits _ {- \pi } ^  \pi  f ^ { 2 } ( t)  dt  = \
 +
 
 +
\frac{a _ {0}  ^ {2} }{2}
 +
+ \sum _ { n= } 1 ^  \infty  ( a _ {n}  ^ {2} + b _ {n}  ^ {2}
 +
),
 +
$$
  
 
which is called the classical Parseval equality. It was proved in 1805 by M. Parseval.
 
which is called the classical Parseval equality. It was proved in 1805 by M. Parseval.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159050.png" /> and
+
If $  g \in L _ {2} [- \pi , \pi ] $
 +
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/p/p071/p071590/p07159051.png" /></td> </tr></table>
+
$$
 +
g  \sim 
 +
\frac{a _ {0}  ^  \prime  }{2}
 +
+ \sum _ { n= } 1 ^  \infty 
 +
( a _ {n}  ^  \prime  \cos  nx + b _ {n}  ^  \prime  \sin  nx ),
 +
$$
  
 
then an equality similar to (2) looks as follows:
 
then an equality similar to (2) looks as follows:
  
<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/p/p071/p071590/p07159052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
  
Two classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159054.png" /> of real-valued functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159055.png" /> and such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159057.png" /> Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071590/p07159061.png" />.
+
\frac{1} \pi
 +
\int\limits _ {- \pi } ^  \pi  f( t) g( t)  dt  = \
 +
 
 +
\frac{1}{2}
 +
a _ {0} a _ {0}  ^  \prime  + \sum _ { n= } 1 ^  \infty  ( a _ {n} a _ {n}  ^  \prime  +
 +
b _ {n} b _ {n}  ^  \prime  ).
 +
$$
 +
 
 +
Two classes  $  K $
 +
and $  K  ^  \prime  $
 +
of real-valued functions defined on $  [- \pi , \pi ] $
 +
and such that for all $  f \in K $
 +
and $  g \in K  ^  \prime  $
 +
Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces $  L _ {p} [- \pi , \pi ] $
 +
and $  L _ {q} [- \pi , \pi ] $,
 +
$  p  ^ {-} 1 + q  ^ {-} 1 = 1 $,  
 +
$  1 < p < + \infty $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Steklov,  "Sur certaines égalités générales communes à plusieurs séries de fonctions souvent employées dans l'analyse"  ''Zap. Nauchn. Fiz.-Mat. Obshch. Ser. 8'' , '''157'''  (1904)  pp. 1–32</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Kirillov,  A.D. Gvishiani,  "Theorems and problems in functional analysis" , Springer  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Steklov,  "Sur certaines égalités générales communes à plusieurs séries de fonctions souvent employées dans l'analyse"  ''Zap. Nauchn. Fiz.-Mat. Obshch. Ser. 8'' , '''157'''  (1904)  pp. 1–32</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Kirillov,  A.D. Gvishiani,  "Theorems and problems in functional analysis" , Springer  (1982)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>

Latest revision as of 08:05, 6 June 2020


An equality expressing the square of the norm of an element in a vector space with a scalar product in terms of the square of the moduli of the Fourier coefficients of this element in some orthogonal system. Thus, if $ X $ is a normed separable vector space with a scalar product $ ( , ) $, if $ \| \cdot \| $ is the corresponding norm and if $ \{ e _ {n} \} $ is an orthogonal system in $ X $, $ e _ {n} \neq 0 $, $ n = 1, 2 \dots $ then Parseval's equality for an element $ x \in X $ is

$$ \tag{1 } \| x \| ^ {2} = \sum _ { n= } 1 ^ \infty | a _ {n} | ^ {2} \| e _ {n} \| ^ {2} , $$

where $ a _ {n} = ( x, e _ {n} )/( e _ {n} , e _ {n} ) $, $ n = 1, 2 \dots $ are the Fourier coefficients of $ x $ in the system $ \{ e _ {n} \} $. If $ \{ e _ {n} \} $ is orthonormal, then Parseval's equality has the form

$$ \| x \| ^ {2} = \sum _ { n= } 1 ^ \infty | a _ {n} | ^ {2} . $$

The validity of Parseval's equality for a given element $ x \in X $ is a necessary and sufficient condition for its Fourier series in the orthogonal system $ \{ e _ {n} \} $ to converge to $ x $ in the norm of $ X $. The validity of Parseval's equality for every element $ x \in X $ is a necessary and sufficient condition for the orthogonal system $ \{ e _ {n} \} $ to be complete in $ X $( cf. Complete system). This implies, in particular, that:

1) if $ X $ is a separable Hilbert space (cf. Hilbert space) and $ \{ e _ {n} \} $ is an orthogonal basis of it, then Parseval's equality holds for $ \{ e _ {n} \} $ for every $ x \in X $;

2) if $ X $ is a separable Hilbert space, $ x , y \in X $, if $ \{ e _ {n} \} $ is an orthonormal basis of $ X $ and if $ a _ {n} = ( x, e _ {n} ) $ and $ b _ {n} = ( y, e _ {n} ) $ are the Fourier coefficients of $ x $ and $ y $, then

$$ \tag{2 } ( x, y) = \sum _ { n= } 1 ^ \infty a _ {n} \overline{ {b _ {n} }}\; , $$

the so-called generalized Parseval equality. In a fairly-definitive form the question of the completeness of a system of functions that are the eigen functions of differential operators was studied by V.A. Steklov in [1].

Parseval's equality can also be generalized to the case of non-separable Hilbert spaces: If $ \{ e _ \alpha \} $, $ \alpha \in \mathfrak A $( $ \mathfrak A $ is a certain index set), is a complete orthonormal system in a Hilbert space $ X $, then for any element $ x \in X $ Parseval's equality holds:

$$ ( x, x) = \sum _ {\alpha \in \mathfrak A } | ( x, e _ \alpha ) | ^ {2} , $$

and the sum on the right-hand side is to be understood as

$$ \sup _ {\mathfrak A _ {0} } \sum _ {\alpha \in \mathfrak A } | ( x, e _ \alpha ) | ^ {2} , $$

where the supremum is taken over all finite subsets $ \mathfrak A _ {0} $ of $ \mathfrak A $.

When $ X = L _ {2} [- \pi , \pi ] $, the space of real-valued functions with Lebesgue-integrable squares on $ [- \pi , \pi ] $, and $ f \in L _ {2} [- \pi , \pi ] $, then one may take the trigonometric system as a complete orthogonal system and

$$ f \sim \frac{a _ {0} }{2} + \sum _ { n= } 1 ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx), $$

where (1) takes the form

$$ \frac{1} \pi \int\limits _ {- \pi } ^ \pi f ^ { 2 } ( t) dt = \ \frac{a _ {0} ^ {2} }{2} + \sum _ { n= } 1 ^ \infty ( a _ {n} ^ {2} + b _ {n} ^ {2} ), $$

which is called the classical Parseval equality. It was proved in 1805 by M. Parseval.

If $ g \in L _ {2} [- \pi , \pi ] $ and

$$ g \sim \frac{a _ {0} ^ \prime }{2} + \sum _ { n= } 1 ^ \infty ( a _ {n} ^ \prime \cos nx + b _ {n} ^ \prime \sin nx ), $$

then an equality similar to (2) looks as follows:

$$ \tag{3 } \frac{1} \pi \int\limits _ {- \pi } ^ \pi f( t) g( t) dt = \ \frac{1}{2} a _ {0} a _ {0} ^ \prime + \sum _ { n= } 1 ^ \infty ( a _ {n} a _ {n} ^ \prime + b _ {n} b _ {n} ^ \prime ). $$

Two classes $ K $ and $ K ^ \prime $ of real-valued functions defined on $ [- \pi , \pi ] $ and such that for all $ f \in K $ and $ g \in K ^ \prime $ Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces $ L _ {p} [- \pi , \pi ] $ and $ L _ {q} [- \pi , \pi ] $, $ p ^ {-} 1 + q ^ {-} 1 = 1 $, $ 1 < p < + \infty $.

References

[1] V.A. Steklov, "Sur certaines égalités générales communes à plusieurs séries de fonctions souvent employées dans l'analyse" Zap. Nauchn. Fiz.-Mat. Obshch. Ser. 8 , 157 (1904) pp. 1–32
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[3] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 2 , MIR (1982) (Translated from Russian)
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[6] A.A. Kirillov, A.D. Gvishiani, "Theorems and problems in functional analysis" , Springer (1982) (Translated from Russian)

Comments

References

[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
How to Cite This Entry:
Parseval equality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parseval_equality&oldid=48131
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article