# 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  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  in the finite-dimensional case, and by J. Moreau  and A. Brøndsted  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  and .

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

How to Cite This Entry:
Conjugate function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_function&oldid=46469
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article