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A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions.
 
A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions.
  
1) The function conjugate to a complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250201.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250202.png" /> whose values are the complex conjugates of those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250203.png" />.
+
1) The function conjugate to a complex-valued function $  f $
 +
is the function $  \overline{f}\; $
 +
whose values are the complex conjugates of those of $  f $.
  
 
2) For the function conjugate to a harmonic function see [[Conjugate harmonic functions|Conjugate harmonic functions]].
 
2) For the function conjugate to a harmonic function see [[Conjugate harmonic functions|Conjugate harmonic functions]].
  
3) The function conjugate to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250205.png" />-periodic summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250207.png" /> is given by
+
3) The function conjugate to a $  2 \pi $-
 +
periodic summable function $  f $
 +
on $  [- \pi , \pi ] $
 +
is given by
 +
 
 +
$$
 +
\overline{f}\; ( x)  = \
 +
\lim\limits _ {\epsilon \rightarrow 0+ } \
 +
- {
 +
\frac{1} \pi
 +
}
 +
\int\limits _  \epsilon  ^  \pi 
 +
 
 +
\frac{f ( x + t) - f ( x - t) }{2  \mathop{\rm tan} ( t / 2) }
 +
  dt;
 +
$$
  
<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/c025/c025020/c0250208.png" /></td> </tr></table>
+
it exists almost-everywhere and coincides almost-everywhere with the  $  ( C, \alpha ) $-
 +
sum,  $  \alpha > 0 $,
 +
and the Abel–Poisson sum of the [[Conjugate trigonometric series|conjugate trigonometric series]].
  
it exists almost-everywhere and coincides almost-everywhere with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c0250209.png" />-sum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502010.png" />, and the Abel–Poisson sum of the [[Conjugate trigonometric series|conjugate trigonometric series]].
+
4) The function conjugate to a function  $  f:  X \rightarrow \overline{\mathbf R}\; $
 +
defined on a vector space  $  X $
 +
dual to a vector space  $  Y $(
 +
with respect to a bilinear form  $  \langle  x, y \rangle $)
 +
is the function on  $  Y $
 +
given by
  
4) The function conjugate to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502011.png" /> defined on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502012.png" /> dual to a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502013.png" /> (with respect to a bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502014.png" />) is the function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502015.png" /> given by
+
$$ \tag{* }
 +
f  ^ {*} ( y) = \
 +
\sup _ {x \in X } \
 +
( \langle  x, y \rangle - f ( x)).
 +
$$
  
<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/c/c025/c025020/c02502016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
The conjugate of a function defined on  $  Y $
 +
is defined in a similar way.
  
The conjugate of a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502017.png" /> is defined in a similar way.
+
The function conjugate to the function $  f _ {p} ( x) = {| x |  ^ {p} } /p $,
 +
$  1 < p < \infty $,
 +
of one variable is given by
  
The function conjugate to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502019.png" />, of one variable is given by
+
$$
 +
f _ {q }  ( y)  = \
  
<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/c025/c025020/c02502020.png" /></td> </tr></table>
+
\frac{| y | ^ {q } }{q }
 +
,\ \
 +
{
 +
\frac{1}{p}
 +
} + {
 +
\frac{1}{q }
 +
= 1.
 +
$$
  
The function conjugate to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502021.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502022.png" /> with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502023.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502024.png" />. The function conjugate to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502025.png" /> on a normed space is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502026.png" /> which is equal to zero when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502027.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502028.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502029.png" />.
+
The function conjugate to the function $  f ( x) = \langle  x, x \rangle/2 $
 +
on a Hilbert space $  X $
 +
with scalar product $  \langle  , \rangle $
 +
is the function $  \langle  y, y \rangle/2 $.  
 +
The function conjugate to the norm $  N ( x) = \| x \| $
 +
on a normed space is the function $  N  ^ {*} ( y) $
 +
which is equal to zero when $  \| y \| < 1 $
 +
and to $  + \infty $
 +
when $  \| y \| > 1 $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502030.png" /> is smooth and increases at infinity faster than any linear function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502031.png" /> is just the [[Legendre transform|Legendre transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502032.png" />. For one-dimensional strictly-convex functions, a definition equivalent to (*) was given by W.H. Young [[#References|[1]]] in other terms. He defined the conjugate of a function
+
If $  f $
 +
is smooth and increases at infinity faster than any linear function, then $  f  ^ {*} $
 +
is just the [[Legendre transform|Legendre transform]] of $  f $.  
 +
For one-dimensional strictly-convex functions, a definition equivalent to (*) was given by W.H. Young [[#References|[1]]] in other terms. He defined the conjugate of a function
  
<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/c025/c025020/c02502033.png" /></td> </tr></table>
+
$$
 +
f ( x)  = \
 +
\int\limits _ { 0 } ^ { x }
 +
\phi ( t)  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502034.png" /> is continuous and strictly increasing, by the relation
+
where $  \phi $
 +
is continuous and strictly increasing, by the relation
  
<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/c025/c025020/c02502035.png" /></td> </tr></table>
+
$$
 +
f  ^ {*} ( y)  = \
 +
\int\limits _ { 0 } ^ { y }
 +
\psi ( t)  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502036.png" /> is the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502037.png" />. Definition (*) was originally proposed by S. Mandelbrojt for one-dimensional functions, by W. Fenchel [[#References|[2]]] in the finite-dimensional case, and by J. Moreau [[#References|[3]]] and A. Brøndsted [[#References|[4]]] in the infinite-dimensional case. For a convex function and its conjugate, Young's inequality holds:
+
where $  \psi $
 +
is the function inverse to $  \phi $.  
 +
Definition (*) was originally proposed by S. Mandelbrojt for one-dimensional functions, by W. Fenchel [[#References|[2]]] in the finite-dimensional case, and by J. Moreau [[#References|[3]]] and A. Brøndsted [[#References|[4]]] in the infinite-dimensional case. For a convex function and its conjugate, Young's inequality 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/c/c025/c025020/c02502038.png" /></td> </tr></table>
+
$$
 +
\langle  x, y \rangle  \leq  \
 +
f ( x) + f  ^ {*} ( y).
 +
$$
  
The conjugate function is a closed convex function. The conjugation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502039.png" /> establishes a one-to-one correspondence between the family of proper closed convex functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502040.png" /> and that of proper closed convex functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502041.png" /> (the Fenchel–Moreau theorem).
+
The conjugate function is a closed convex function. The conjugation operator $  *: f \mapsto f  ^ {*} $
 +
establishes a one-to-one correspondence between the family of proper closed convex functions on $  X $
 +
and that of proper closed convex functions on $  Y $(
 +
the Fenchel–Moreau theorem).
  
 
For more details see [[#References|[5]]] and [[#References|[6]]].
 
For more details see [[#References|[5]]] and [[#References|[6]]].
Line 43: Line 122:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.H. Young,  "On classes of summable functions and their Fourier series"  ''Proc. Roy. Soc. Ser. A.'' , '''87'''  (1912)  pp. 225–229  {{MR|}}  {{ZBL|43.1114.12}}  {{ZBL|43.0334.09}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77  {{MR|0028365}} {{ZBL|0038.20902}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.J. Moreau,  "Fonctions convexes en dualité" , Univ. Montpellier  (1962)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Brøndsted,  "Conjugate convex functions in topological vector spaces"  ''Math. Fys. Medd. Danske vid. Selsk.'' , '''34''' :  2  (1964)  pp. 1–26  {{MR|}} {{ZBL|0119.10004}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)  {{MR|0274683}} {{ZBL|0193.18401}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.M. Alekseev,  V.M. Tikhomirov,  S.V. Fomin,  "Commande optimale" , MIR  (1982)  (Translated from Russian)  {{MR|728225}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.H. Young,  "On classes of summable functions and their Fourier series"  ''Proc. Roy. Soc. Ser. A.'' , '''87'''  (1912)  pp. 225–229  {{MR|}}  {{ZBL|43.1114.12}}  {{ZBL|43.0334.09}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77  {{MR|0028365}} {{ZBL|0038.20902}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.J. Moreau,  "Fonctions convexes en dualité" , Univ. Montpellier  (1962)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Brøndsted,  "Conjugate convex functions in topological vector spaces"  ''Math. Fys. Medd. Danske vid. Selsk.'' , '''34''' :  2  (1964)  pp. 1–26  {{MR|}} {{ZBL|0119.10004}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)  {{MR|0274683}} {{ZBL|0193.18401}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.M. Alekseev,  V.M. Tikhomirov,  S.V. Fomin,  "Commande optimale" , MIR  (1982)  (Translated from Russian)  {{MR|728225}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The concepts of conjugate harmonic functions and conjugate trigonometric series are not unrelated. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502042.png" /> be a harmonic function on the closed unit disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502043.png" /> its harmonic conjugate, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502046.png" /> is the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502047.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502048.png" /> be the boundary value function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502049.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502050.png" />. Then one has the Poisson integral representation
+
The concepts of conjugate harmonic functions and conjugate trigonometric series are not unrelated. Let $  u $
 +
be a harmonic function on the closed unit disc and $  \widetilde{u}  $
 +
its harmonic conjugate, so that $  u = \mathop{\rm Re} ( \phi ) $,  
 +
$  \widetilde{u}  = \mathop{\rm Im} ( \phi ) $,  
 +
where $  \phi $
 +
is the analytic function $  u + i \widetilde{u}  $.  
 +
Let $  g ( t) $
 +
be the boundary value function of $  u $,  
 +
i.e. $  g ( t) = u ( e  ^ {it} ) $.  
 +
Then one has the Poisson integral representation
  
<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/c025/c025020/c02502051.png" /></td> </tr></table>
+
$$
 +
u ( re ^ {i \theta } )  = \int\limits _ {- \pi } ^  \pi  P _ {r} ( \theta -
 +
t) g ( t)  dt ,
 +
$$
  
 
where
 
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/c/c025/c025020/c02502052.png" /></td> </tr></table>
+
$$
 +
P _ {r} ( s) =  
 +
\frac{1}{2 \pi }
 +
  \mathop{\rm Re} 
 +
\frac{1 + re
 +
^ {is} }{1 - re  ^ {is} }
 +
,
 +
$$
  
 
and
 
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/c/c025/c025020/c02502053.png" /></td> </tr></table>
+
$$
 +
\widetilde{u}  ( re ^ {i \theta } )  = \
 +
\int\limits _ {- \pi } ^  \pi  Q _ {r} ( \theta - t) g ( t)  dt ,
 +
$$
  
 
with
 
with
  
<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/c025/c025020/c02502054.png" /></td> </tr></table>
+
$$
 +
Q _ {r} = {
 +
\frac{1}{2 \pi }
 +
}  \mathop{\rm Im} 
 +
\frac{1 + re  ^ {is} }{1 - re  ^ {is} }
 +
.
 +
$$
  
Then letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502055.png" />, (formally)
+
Then letting $  r \uparrow 1 $,  
 +
(formally)
  
<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/c025/c025020/c02502056.png" /></td> </tr></table>
+
$$
 +
\widetilde{u}  ( e ^ {i
 +
\theta } )  = {
 +
\frac{1}{2 \pi }
 +
} \int\limits _ { 0 } ^  \pi 
 +
\frac{g ( \theta - t) - g
 +
( \theta + t) }{ \mathop{\rm tan} ( t / 2) }
 +
  dt
 +
$$
  
is precisely the conjugate trigonometric series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025020/c02502057.png" />.
+
is precisely the conjugate trigonometric series of $  g ( t) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1959)  {{MR|0107776}} {{ZBL|0085.05601}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1959)  {{MR|0107776}} {{ZBL|0085.05601}} </TD></TR></table>

Revision as of 17:46, 4 June 2020


A concept in the theory of functions which is a concrete image of some involutory operator for the corresponding class of functions.

1) The function conjugate to a complex-valued function $ f $ is the function $ \overline{f}\; $ whose values are the complex conjugates of those of $ f $.

2) For the function conjugate to a harmonic function see Conjugate harmonic functions.

3) The function conjugate to a $ 2 \pi $- periodic summable function $ f $ on $ [- \pi , \pi ] $ is given by

$$ \overline{f}\; ( x) = \ \lim\limits _ {\epsilon \rightarrow 0+ } \ - { \frac{1} \pi } \int\limits _ \epsilon ^ \pi \frac{f ( x + t) - f ( x - t) }{2 \mathop{\rm tan} ( t / 2) } dt; $$

it exists almost-everywhere and coincides almost-everywhere with the $ ( C, \alpha ) $- sum, $ \alpha > 0 $, and the Abel–Poisson sum of the conjugate trigonometric series.

4) The function conjugate to a function $ f: X \rightarrow \overline{\mathbf R}\; $ defined on a vector space $ X $ dual to a vector space $ Y $( with respect to a bilinear form $ \langle x, y \rangle $) is the function on $ Y $ given by

$$ \tag{* } f ^ {*} ( y) = \ \sup _ {x \in X } \ ( \langle x, y \rangle - f ( x)). $$

The conjugate of a function defined on $ Y $ is defined in a similar way.

The function conjugate to the function $ f _ {p} ( x) = {| x | ^ {p} } /p $, $ 1 < p < \infty $, of one variable is given by

$$ f _ {q } ( y) = \ \frac{| y | ^ {q } }{q } ,\ \ { \frac{1}{p} } + { \frac{1}{q } } = 1. $$

The function conjugate to the function $ f ( x) = \langle x, x \rangle/2 $ on a Hilbert space $ X $ with scalar product $ \langle , \rangle $ is the function $ \langle y, y \rangle/2 $. The function conjugate to the norm $ N ( x) = \| x \| $ on a normed space is the function $ N ^ {*} ( y) $ which is equal to zero when $ \| y \| < 1 $ and to $ + \infty $ when $ \| y \| > 1 $.

If $ f $ is smooth and increases at infinity faster than any linear function, then $ f ^ {*} $ is just the Legendre transform of $ f $. For one-dimensional strictly-convex functions, a definition equivalent to (*) was given by W.H. Young [1] in other terms. He defined the conjugate of a function

$$ f ( x) = \ \int\limits _ { 0 } ^ { x } \phi ( t) dt, $$

where $ \phi $ is continuous and strictly increasing, by the relation

$$ f ^ {*} ( y) = \ \int\limits _ { 0 } ^ { y } \psi ( t) dt, $$

where $ \psi $ is the function inverse to $ \phi $. Definition (*) was originally proposed by S. Mandelbrojt for one-dimensional functions, by W. Fenchel [2] in the finite-dimensional case, and by J. Moreau [3] and A. Brøndsted [4] in the infinite-dimensional case. For a convex function and its conjugate, Young's inequality holds:

$$ \langle x, y \rangle \leq \ f ( x) + f ^ {*} ( y). $$

The conjugate function is a closed convex function. The conjugation operator $ *: f \mapsto f ^ {*} $ establishes a one-to-one correspondence between the family of proper closed convex functions on $ X $ and that of proper closed convex functions on $ Y $( the Fenchel–Moreau theorem).

For more details see [5] and [6].

See also Convex analysis; Support function; Duality in extremal problems, Convex analysis; Dual functions.

References

[1] W.H. Young, "On classes of summable functions and their Fourier series" Proc. Roy. Soc. Ser. A. , 87 (1912) pp. 225–229 Zbl 43.1114.12 Zbl 43.0334.09
[2] W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77 MR0028365 Zbl 0038.20902
[3] J.J. Moreau, "Fonctions convexes en dualité" , Univ. Montpellier (1962)
[4] A. Brøndsted, "Conjugate convex functions in topological vector spaces" Math. Fys. Medd. Danske vid. Selsk. , 34 : 2 (1964) pp. 1–26 Zbl 0119.10004
[5] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Zbl 0193.18401
[6] V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, "Commande optimale" , MIR (1982) (Translated from Russian) MR728225

Comments

The concepts of conjugate harmonic functions and conjugate trigonometric series are not unrelated. Let $ u $ be a harmonic function on the closed unit disc and $ \widetilde{u} $ its harmonic conjugate, so that $ u = \mathop{\rm Re} ( \phi ) $, $ \widetilde{u} = \mathop{\rm Im} ( \phi ) $, where $ \phi $ is the analytic function $ u + i \widetilde{u} $. Let $ g ( t) $ be the boundary value function of $ u $, i.e. $ g ( t) = u ( e ^ {it} ) $. Then one has the Poisson integral representation

$$ u ( re ^ {i \theta } ) = \int\limits _ {- \pi } ^ \pi P _ {r} ( \theta - t) g ( t) dt , $$

where

$$ P _ {r} ( s) = \frac{1}{2 \pi } \mathop{\rm Re} \frac{1 + re ^ {is} }{1 - re ^ {is} } , $$

and

$$ \widetilde{u} ( re ^ {i \theta } ) = \ \int\limits _ {- \pi } ^ \pi Q _ {r} ( \theta - t) g ( t) dt , $$

with

$$ Q _ {r} = { \frac{1}{2 \pi } } \mathop{\rm Im} \frac{1 + re ^ {is} }{1 - re ^ {is} } . $$

Then letting $ r \uparrow 1 $, (formally)

$$ \widetilde{u} ( e ^ {i \theta } ) = { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ \pi \frac{g ( \theta - t) - g ( \theta + t) }{ \mathop{\rm tan} ( t / 2) } dt $$

is precisely the conjugate trigonometric series of $ g ( t) $.

References

[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1959) MR0107776 Zbl 0085.05601
How to Cite This Entry:
Conjugate function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_function&oldid=28162
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article