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