# Conjugate function

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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  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  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: 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  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=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