View source for Parseval equality

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

<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>

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

<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>

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:

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" />;

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

<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>

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:

<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>

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>

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" />.

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

<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>

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>

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

<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>

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>

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" />.

====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>



====Comments====


====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>