代表的な凸共役の表
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/06/20 04:59 UTC 版)
次の表では、多くの有名な函数のルジャンドル変換で、有用な性質を持つものが示されている。 g ( x ) {\displaystyle g(x)} dom ( g ) {\displaystyle \operatorname {dom} (g)} g ∗ ( x ∗ ) {\displaystyle g^{*}(x^{*})} dom ( g ∗ ) {\displaystyle \operatorname {dom} (g^{*})} f ( a x ) {\displaystyle f(ax)} (where a ≠ 0 {\displaystyle a\neq 0} ) X {\displaystyle X} f ∗ ( x ∗ a ) {\displaystyle f^{*}\left({\frac {x^{*}}{a}}\right)} X ∗ {\displaystyle X^{*}} f ( x + b ) {\displaystyle f(x+b)} X {\displaystyle X} f ∗ ( x ∗ ) − ⟨ b , x ∗ ⟩ {\displaystyle f^{*}(x^{*})-\langle b,x^{*}\rangle } X ∗ {\displaystyle X^{*}} a f ( x ) {\displaystyle af(x)} (where a > 0 {\displaystyle a>0} ) X {\displaystyle X} a f ∗ ( x ∗ a ) {\displaystyle af^{*}\left({\frac {x^{*}}{a}}\right)} X ∗ {\displaystyle X^{*}} α + β x + γ ⋅ f ( λ x + δ ) {\displaystyle \alpha +\beta x+\gamma \cdot f(\lambda x+\delta )} X {\displaystyle X} − α − δ x ∗ − β λ + γ ⋅ f ∗ ( x ∗ − β γ λ ) ( γ > 0 ) {\displaystyle -\alpha -\delta {\frac {x^{*}-\beta }{\lambda }}+\gamma \cdot f^{*}\left({\frac {x^{*}-\beta }{\gamma \lambda }}\right)\quad (\gamma >0)} X ∗ {\displaystyle X^{*}} | x | p p {\displaystyle {\frac {|x|^{p}}{p}}} (where p > 1 {\displaystyle p>1} ) R {\displaystyle \mathbb {R} } | x ∗ | q q {\displaystyle {\frac {|x^{*}|^{q}}{q}}} (where 1 p + 1 q = 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1} ) R {\displaystyle \mathbb {R} } − x p p {\displaystyle {\frac {-x^{p}}{p}}} (where 0 < p < 1 {\displaystyle 0<p<1} ) R + {\displaystyle \mathbb {R} _{+}} − ( − x ∗ ) q q {\displaystyle {\frac {-(-x^{*})^{q}}{q}}} (where 1 p + 1 q = 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1} ) R − − {\displaystyle \mathbb {R} _{--}} 1 + x 2 {\displaystyle {\sqrt {1+x^{2}}}} R {\displaystyle \mathbb {R} } − 1 − ( x ∗ ) 2 {\displaystyle -{\sqrt {1-(x^{*})^{2}}}} [ − 1 , 1 ] {\displaystyle [-1,1]} − log ( x ) {\displaystyle -\log(x)} R + + {\displaystyle \mathbb {R} _{++}} − ( 1 + log ( − x ∗ ) ) {\displaystyle -(1+\log(-x^{*}))} R − − {\displaystyle \mathbb {R} _{--}} e x {\displaystyle e^{x}} R {\displaystyle \mathbb {R} } { x ∗ log ( x ∗ ) − x ∗ if x ∗ > 0 0 if x ∗ = 0 {\displaystyle {\begin{cases}x^{*}\log(x^{*})-x^{*}&{\text{if }}x^{*}>0\\0&{\text{if }}x^{*}=0\end{cases}}} R + {\displaystyle \mathbb {R} _{+}} log ( 1 + e x ) {\displaystyle \log \left(1+e^{x}\right)} R {\displaystyle \mathbb {R} } { x ∗ log ( x ∗ ) + ( 1 − x ∗ ) log ( 1 − x ∗ ) if 0 < x ∗ < 1 0 if x ∗ = 0 , 1 {\displaystyle {\begin{cases}x^{*}\log(x^{*})+(1-x^{*})\log(1-x^{*})&{\text{if }}0<x^{*}<1\\0&{\text{if }}x^{*}=0,1\end{cases}}} [ 0 , 1 ] {\displaystyle [0,1]} − log ( 1 − e x ) {\displaystyle -\log \left(1-e^{x}\right)} R {\displaystyle \mathbb {R} } { x ∗ log ( x ∗ ) − ( 1 + x ∗ ) log ( 1 + x ∗ ) if x ∗ > 0 0 if x ∗ = 0 {\displaystyle {\begin{cases}x^{*}\log(x^{*})-(1+x^{*})\log(1+x^{*})&{\text{if }}x^{*}>0\\0&{\text{if }}x^{*}=0\end{cases}}} R + {\displaystyle \mathbb {R} _{+}}
※この「代表的な凸共役の表」の解説は、「凸共役性」の解説の一部です。
「代表的な凸共役の表」を含む「凸共役性」の記事については、「凸共役性」の概要を参照ください。
- 代表的な凸共役の表のページへのリンク
