Dual functions

Functions complementary in the sense of Young, i.e. strictly convex functions (cf. Convex function (of a real variable)) connected by the Legendre transform.

Comments

For certain real-valued non-decreasing functions defined on the positive half-line (including zero) there is a natural notion of an inverse. If and are such inverses to each other, the functions and defined (on the positive half-line) by

are said to be complementary in the sense of Young or Young-conjugate. For them Young's inequality holds:

Associated with a pair of non-vanishing functions complementary in the sense of Young and a -finite measure, there is a pair , of complete normed spaces. These spaces, consisting of (equivalence classes of) -measurable functions, are called Orlicz spaces (cf. Orlicz space). The Lebesgue spaces (cf. Lebesgue space) are particular cases of Orlicz spaces, cf. [a4].

In a more abstract setting, the name dual functions is reminiscent of dual pair in duality theory and of dual problems in convex programming and optimal control (cf. Optimal control, mathematical theory of), but this name is rarely used in English: the most common name is (convex) conjugate functions (cf. Conjugate function).

Let and be two real vector spaces in separate duality with respect to a bilinear form (the usual one if ), and let be a mapping from into (if is only defined on a subset of , set on , the complement of ). If is non-empty, the dual, or polar, or adjoint, or better conjugate function of is the convex function defined on by

The following result is a generalization of the geometric Hahn–Banach theorem on the bipolar of a set: the biconjugate function of is the greatest lower semi-continuous convex function bounded above by , and so is equal to if and only if is a lower semi-continuous convex function (in which case is called a pair of conjugate functions). The notion of conjugate function, which was introduced by W. Young in the case and by W. Fenchel in the case , is very important in convex analysis; it is closely related to the notion of subdifferential: if is convex and is its subdifferential, then for and one has

This can be written, if is lower semi-continuous, as

The mapping is often called the Fenchel transform, sometimes with the name of Young or Legendre, or both, added. When and is sufficiently smooth as a convex function, it is a special case of the Legendre transform; on the other hand, it is also a special case of a Galois correspondence; these facts are of secondary importance in convex analysis. The notion of conjugate function plays a fundamental role in convex optimization. It is used to define the Lagrangian of some problem and the associated dual problem.

When , a function is called a Young function if it is a non-decreasing convex function such that and . The conjugate function of a Young function is still a Young function on ; for example, when with , then where is the conjugate exponent of , i.e. . Young functions are used to define Orlicz spaces (cf. Orlicz space), and pairs of conjugate Young functions are used to study the duality between them; more generally they help to establish various inequalities in measure theory (Burkholder inequalities in martingale theory, Chernov's inequality in classical probability theory, Kullback's inequality in statistics, etc.), via the easy to prove but fundamental Young inequality

which enabled Young to solve a problem about Fourier transformation.

References

[a1]	R.T. Rockafellar, "Conjugate duality and optimization" , Reg. Conf. Ser. Appl. Math. , SIAM (1974)
[a2]	J. Neveu, "Martingales à temps discret" , Masson (1972)
[a3]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 2. Theory of martingales , North-Holland (1978–1988) (Translated from French)
[a4]	A.C. Zaanen, "Linear analysis" , North-Holland (1956)