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Difference between revisions of "Conjugate function"

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====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</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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.J. Moreau,  "Fonctions convexes en dualité" , Univ. Montpellier  (1962)</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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</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)</TD></TR></table>
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<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>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1959)</TD></TR></table>
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<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 11:58, 27 September 2012

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 is the function whose values are the complex conjugates of those of .

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

3) The function conjugate to a -periodic summable function on is given by

it exists almost-everywhere and coincides almost-everywhere with the -sum, , and the Abel–Poisson sum of the conjugate trigonometric series.

4) The function conjugate to a function defined on a vector space dual to a vector space (with respect to a bilinear form ) is the function on given by

(*)

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

The function conjugate to the function , , of one variable is given by

The function conjugate to the function on a Hilbert space with scalar product is the function . The function conjugate to the norm on a normed space is the function which is equal to zero when and to when .

If is smooth and increases at infinity faster than any linear function, then is just the Legendre transform of . 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

where is continuous and strictly increasing, by the relation

where is the function inverse to . 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:

The conjugate function is a closed convex function. The conjugation operator establishes a one-to-one correspondence between the family of proper closed convex functions on and that of proper closed convex functions on (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 be a harmonic function on the closed unit disc and its harmonic conjugate, so that , , where is the analytic function . Let be the boundary value function of , i.e. . Then one has the Poisson integral representation

where

and

with

Then letting , (formally)

is precisely the conjugate trigonometric series of .

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=13183
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article