Namespaces
Variants
Actions

Difference between revisions of "Conjugate class of functions"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A concept in the theory of functions which is a concrete instance of [[Duality|duality]] in functions spaces. Thus, if a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249901.png" /> of functions is regarded as a Banach space or a topological vector space, then the conjugate class is defined as the class of functions isometrically isomorphic to the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249902.png" />. For example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249904.png" />, there is an isometric isomorphism between the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249906.png" />, under which corresponding elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c0249908.png" /> are related by
+
<!--
 +
c0249901.png
 +
$#A+1 = 22 n = 0
 +
$#C+1 = 22 : ~/encyclopedia/old_files/data/C024/C.0204990 Conjugate class of functions
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/c/c024/c024990/c0249909.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
If one considers some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499011.png" />-periodic summable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499012.png" />, then the conjugate class is defined to be the class of functions conjugate to the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499013.png" />. For example, the class conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499015.png" />) coincides with the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499017.png" /> for which
+
A concept in the theory of functions which is a concrete instance of [[Duality|duality]] in functions spaces. Thus, if a class $  X $
 +
of functions is regarded as a Banach space or a topological vector space, then the conjugate class is defined as the class of functions isometrically isomorphic to the dual space  $  X  ^ {*} $.  
 +
For example, when  $  1 \leq  p < \infty $
 +
and  $  1 / p + 1 / q = 1 $,
 +
there is an isometric isomorphism between the spaces  $  ( L _ {p} [ a , b ] ) ^ {*} $
 +
and  $  L _ {q} [ a , b ] $,
 +
under which corresponding elements  $  x  ^ {*} $
 +
and  $  g $
 +
are related by
  
<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/c/c024/c024990/c02499018.png" /></td> </tr></table>
+
$$
 +
x  ^ {*} ( f  )  = \
 +
\int\limits _ { a } ^ { b }
 +
g ( x) f ( x)  d x .
 +
$$
  
The class conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499020.png" />, coincides with the class of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024990/c02499022.png" />.
+
If one considers some class  $  X $
 +
of  $  2 \pi $-
 +
periodic summable functions on  $  [ - \pi , \pi ] $,
 +
then the conjugate class is defined to be the class of functions conjugate to the functions in  $  X $.
 +
For example, the class conjugate to $  L _ {p} [ - \pi , \pi ] $(
 +
$  1 < p < \infty $)
 +
coincides with the class of functions  $  f $
 +
in  $  L _ {p} [ - \pi , \pi ] $
 +
for which
 +
 
 +
$$
 +
\int\limits _ {- \pi } ^  \pi 
 +
f ( x)  d x  = 0 .
 +
$$
 +
 
 +
The class conjugate to  $  \mathop{\rm Lip}  \alpha $,  
 +
$  0 < \alpha < 1 $,  
 +
coincides with the class of functions in $  \mathop{\rm Lip}  \alpha $
 +
for which $  \int _ {- \pi }  ^  \pi  f ( x)  d x = 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Fréchet,  ''C.R. Acad. Sci.'' , '''144'''  (1907)  pp. 1414–1416</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  ''C.R. Acad. Sci.'' , '''144'''  (1907)  pp. 1409–1411</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. [I. Privalov] Priwaloff,  ''Bull. Soc. Math. France'' , '''44'''  (1916)  pp. 100–103</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">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Fréchet,  ''C.R. Acad. Sci.'' , '''144'''  (1907)  pp. 1414–1416</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  ''C.R. Acad. Sci.'' , '''144'''  (1907)  pp. 1409–1411</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. [I. Privalov] Priwaloff,  ''Bull. Soc. Math. France'' , '''44'''  (1916)  pp. 100–103</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">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


A concept in the theory of functions which is a concrete instance of duality in functions spaces. Thus, if a class $ X $ of functions is regarded as a Banach space or a topological vector space, then the conjugate class is defined as the class of functions isometrically isomorphic to the dual space $ X ^ {*} $. For example, when $ 1 \leq p < \infty $ and $ 1 / p + 1 / q = 1 $, there is an isometric isomorphism between the spaces $ ( L _ {p} [ a , b ] ) ^ {*} $ and $ L _ {q} [ a , b ] $, under which corresponding elements $ x ^ {*} $ and $ g $ are related by

$$ x ^ {*} ( f ) = \ \int\limits _ { a } ^ { b } g ( x) f ( x) d x . $$

If one considers some class $ X $ of $ 2 \pi $- periodic summable functions on $ [ - \pi , \pi ] $, then the conjugate class is defined to be the class of functions conjugate to the functions in $ X $. For example, the class conjugate to $ L _ {p} [ - \pi , \pi ] $( $ 1 < p < \infty $) coincides with the class of functions $ f $ in $ L _ {p} [ - \pi , \pi ] $ for which

$$ \int\limits _ {- \pi } ^ \pi f ( x) d x = 0 . $$

The class conjugate to $ \mathop{\rm Lip} \alpha $, $ 0 < \alpha < 1 $, coincides with the class of functions in $ \mathop{\rm Lip} \alpha $ for which $ \int _ {- \pi } ^ \pi f ( x) d x = 0 $.

References

[1] M. Fréchet, C.R. Acad. Sci. , 144 (1907) pp. 1414–1416
[2] F. Riesz, C.R. Acad. Sci. , 144 (1907) pp. 1409–1411
[3] I. [I. Privalov] Priwaloff, Bull. Soc. Math. France , 44 (1916) pp. 100–103
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Conjugate class of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_class_of_functions&oldid=46467
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article