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 MR Template:ZBL Template:ZBL |
| [2] | W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77 MR0028365 Template:ZBL |
| [3] | J.J. Moreau, "Fonctions convexes en dualité" , Univ. Montpellier (1962) MR Template:ZBL |
| [4] | A. Brøndsted, "Conjugate convex functions in topological vector spaces" Math. Fys. Medd. Danske vid. Selsk. , 34 : 2 (1964) pp. 1–26 MR Template:ZBL |
| [5] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Template:ZBL |
| [6] | V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, "Commande optimale" , MIR (1982) (Translated from Russian) MR728225 Template:ZBL |
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 Template:ZBL |
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