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 $ 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