Conjugate function

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