# लजान्द्र रूपान्तर

लजान्द्र रूपान्तर की ज्यामितीय व्याख्या

गणित में किसी वास्तविक मान वाले, तथा सभी बिन्दुओं पर अवकलनीय फलन f तथा g में निम्नलिखित सम्बन्ध हो तो g को f का लजान्द्र रूपान्तर (LegendreTransform) कहा जाता है। इस रूपान्तर का नाम फ्रांसीसी गणितज्ञ आद्रियें मारि लजान्द्र (Adrien-Marie Legendre) के नाम पर पड़ा है।

${\displaystyle Df=\left(Dg\right)^{-1}}$

जहाँ D , अवकलज (डिफरेंशियल) का प्रतीक है तथा दाहिनी ओर आने वाला -1 , प्रतिलोम फलन को सूचित कर रहा है। यह आसानी से दिखाया जा सकता है कि g , f का लजान्द्र रूपान्तर हो तो f, g का लजान्द्र रूपान्तर होगा

उदाहरण के लिये, फलन ${\displaystyle f(x)={\tfrac {x^{2}}{2}}}$ तथा फलन ${\displaystyle g(p)=-{\tfrac {p^{2}}{2}}}$ एक दूसरे के लजान्द्र रूपान्तर हैं।

एक विशेष स्थिति में, यदि फलन f एक उत्तल फलन (कान्वेक्स फंक्शन) हो तो इसका लजान्द्र रूपान्तर ƒ* निम्नलिखित सम्बन्ध द्वारा अभिव्यक्त किया जा सकता है-

${\displaystyle f^{\star }(p)=\sup _{x}{\bigl (}px-f(x){\bigr )}.}$

## उदाहरण

### प्रथम उदाहरण

Let f(x) = cx2 defined on R, where c > 0 is a fixed constant.

For x* fixed, the function x*xf(x) = x*xcx2 of x has the first derivative x* – 2cx and second derivative −2c; there is one stationary point at x = x*/2c, which is always a maximum. Thus, I* = R and

${\displaystyle f^{*}(x^{*})={\frac {{x^{*}}^{2}}{4c}}}$

Clearly,

${\displaystyle f^{**}(x)={\frac {1}{4(1/4c)}}x^{2}=cx^{2},}$

namely f ** = f.

### द्वितीय उदाहरण

Let f(x) = x2 for xI = [2, 3].

For x* fixed, x*xf(x) is continuous on I compact, hence it always takes a finite maximum on it; it follows that I* = R. The stationary point at x = x*/2 is in the domain [2, 3] if and only if 4 ≤ x* ≤ 6, otherwise the maximum is taken either at x = 2, or x = 3. It follows that

${\displaystyle f^{*}(x^{*})={\begin{cases}2x^{*}-4,\quad &x^{*}<4\\{\frac {{x^{*}}^{2}}{4}},&4\leqslant x^{*}\leqslant 6,\\3x^{*}-9,&x^{*}>6\end{cases}}}$ .

### तृतीय उदाहरण

The function f(x) = cx is convex, for every x (strict convexity is not required for the Legendre transformation to be well defined). Clearly x*xf(x) = (x* − c)x is never bounded from above as a function of x, unless x* − c = 0. Hence f* is defined on I* = {c} and f*(c) = 0.

One may check involutivity: of course x*xf*(x*) is always bounded as a function of x* ∈ {c}, hence I ** = R. Then, for all x one has

${\displaystyle \sup _{x^{*}\in \{c\}}(xx^{*}-f^{*}(x^{*}))=xc,}$

and hence f **(x) = cx = f(x).

### चतुर्थ उदाहरण (अनेक चर)

Let

${\displaystyle f(x)=\langle x,Ax\rangle +c}$

be defined on X = Rn, where A is a real, positive definite matrix. Then f is convex, and

${\displaystyle \langle p,x\rangle -f(x)=\langle p,x\rangle -\langle x,Ax\rangle -c,}$

has gradient p − 2Ax and Hessian −2A, which is negative; hence the stationary point x = A−1p/2 is a maximum. We have X* = Rn, and

${\displaystyle f^{*}(p)={\frac {1}{4}}\langle p,A^{-1}p\rangle -c}$ .

## गुण

${\displaystyle f(x)}$ ${\displaystyle \operatorname {dom} f}$ ${\displaystyle f^{\star }(x^{\star })}$ ${\displaystyle \operatorname {dom} f^{\star }}$ शर्तें
${\displaystyle af(x)}$ ${\displaystyle \operatorname {dom} f}$ ${\displaystyle af^{\star }(x^{\star }/a)}$ ${\displaystyle a\cdot \operatorname {dom} f^{\star }}$ ${\displaystyle a>0}$
${\displaystyle f(ax)}$ ${\displaystyle a^{-1}\cdot \operatorname {dom} f}$ ${\displaystyle f^{\star }(x^{\star }/a)}$ ${\displaystyle a\cdot \operatorname {dom} f^{\star }}$ ${\displaystyle a>0}$
${\displaystyle f(x)+a}$ ${\displaystyle \operatorname {dom} f}$ ${\displaystyle f^{\star }(x^{\star })-a}$ ${\displaystyle \operatorname {dom} f^{\star }}$ ${\displaystyle a\in \mathbb {R} }$
${\displaystyle f(x-a)}$ ${\displaystyle a+\operatorname {dom} f}$ ${\displaystyle f^{\star }(x^{\star })+ax^{\star }}$ ${\displaystyle \operatorname {dom} f^{\star }}$ ${\displaystyle a\in \mathbb {R} }$
${\displaystyle f(x)+ax}$ ${\displaystyle \operatorname {dom} f}$ ${\displaystyle f^{\star }(x^{\star }-a)}$ ${\displaystyle a+\operatorname {dom} f^{\star }}$ ${\displaystyle a\in \mathbb {R} }$
${\displaystyle f(x)+g(x)}$ ${\displaystyle \operatorname {dom} f\cap \operatorname {dom} g}$ ${\displaystyle (f^{\star }\star _{\text{inf}}g^{\star })(x^{\star })}$ ${\displaystyle \operatorname {dom} f^{\star }+\operatorname {dom} g^{\star }}$ ${\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}}$
${\displaystyle (f\star _{\text{inf}}g)(x)}$ ${\displaystyle \operatorname {dom} f+\operatorname {dom} g}$ ${\displaystyle f^{\star }(x^{\star })+g^{\star }(x^{\star })}$ ${\displaystyle \operatorname {dom} f^{\star }\cap \operatorname {dom} g^{\star }}$ ${\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}}$
${\displaystyle ax+b}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle -b}$ ${\displaystyle \{a\}}$
${\displaystyle |x|^{p}/p}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle |x^{\star }|^{p^{\star }}/p^{\star }}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle 1/p+1/p^{\star }=1}$, ${\displaystyle p>1}$
${\displaystyle -x^{p}/p}$ ${\displaystyle [0,\infty )}$ ${\displaystyle -|x^{\star }|^{p^{\star }}/p^{\star }}$ ${\displaystyle (-\infty ,0]}$ ${\displaystyle 1/p+1/p^{\star }=1}$, ${\displaystyle p<1}$
${\displaystyle \exp(x)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x^{\star }(\ln(x^{\star })-1)}$ ${\displaystyle \mathbb {R} ^{+}}$
${\displaystyle x\ln(x)}$ ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle \exp(x-1)}$ ${\displaystyle \mathbb {R} }$
${\displaystyle -1/2-\ln x}$ ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle -1/2-\ln |x^{\star }|}$ ${\displaystyle \mathbb {R} ^{-}}$
${\displaystyle x\exp(x+1)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x^{\star }(W(x^{\star })-1)^{2}/W(x^{\star })}$ ${\displaystyle [-1/e,\infty )}$ लैम्बर्ट का W फलन