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Legendre transform (integral transform)

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This article is about an integral transform using Legendre polynomials. For the involution transform commonly used in classical mechanics and thermodynamics, see Legendre transformation.

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P n ( x ) {\displaystyle P_{n}(x)} {\displaystyle P_{n}(x)} as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} is[1] [2] [3]

J n { f ( x ) } = f ~ ( n ) = ∫ − 1 1 P n ( x )   f ( x )   d x {\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx} {\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx}

The inverse Legendre transform is given by

J n − 1 { f ~ ( n ) } = f ( x ) = ∑ n = 0 ∞ 2 n + 1 2 f ~ ( n ) P n ( x ) {\displaystyle {\mathcal {J}}_{n}^{-1}\{{\tilde {f}}(n)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\tilde {f}}(n)P_{n}(x)} {\displaystyle {\mathcal {J}}_{n}^{-1}\{{\tilde {f}}(n)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\tilde {f}}(n)P_{n}(x)}

Associated Legendre transform

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Associated Legendre transform is defined as

J n , m { f ( x ) } = f ~ ( n , m ) = ∫ − 1 1 ( 1 − x 2 ) − m / 2 P n m ( x )   f ( x )   d x {\displaystyle {\mathcal {J}}_{n,m}\{f(x)\}={\tilde {f}}(n,m)=\int _{-1}^{1}(1-x^{2})^{-m/2}P_{n}^{m}(x)\ f(x)\ dx} {\displaystyle {\mathcal {J}}_{n,m}\{f(x)\}={\tilde {f}}(n,m)=\int _{-1}^{1}(1-x^{2})^{-m/2}P_{n}^{m}(x)\ f(x)\ dx}

The inverse Legendre transform is given by

J n , m − 1 { f ~ ( n , m ) } = f ( x ) = ∑ n = 0 ∞ 2 n + 1 2 ( n − m ) ! ( n + m ) ! f ~ ( n , m ) ( 1 − x 2 ) m / 2 P n m ( x ) {\displaystyle {\mathcal {J}}_{n,m}^{-1}\{{\tilde {f}}(n,m)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\frac {(n-m)!}{(n+m)!}}{\tilde {f}}(n,m)(1-x^{2})^{m/2}P_{n}^{m}(x)} {\displaystyle {\mathcal {J}}_{n,m}^{-1}\{{\tilde {f}}(n,m)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\frac {(n-m)!}{(n+m)!}}{\tilde {f}}(n,m)(1-x^{2})^{m/2}P_{n}^{m}(x)}

Some Legendre transform pairs

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f ( x ) {\displaystyle f(x),円} {\displaystyle f(x),円} f ~ ( n ) {\displaystyle {\tilde {f}}(n),円} {\displaystyle {\tilde {f}}(n),円}
x n {\displaystyle x^{n},円} {\displaystyle x^{n},円} 2 n + 1 ( n ! ) 2 ( 2 n + 1 ) ! {\displaystyle {\frac {2^{n+1}(n!)^{2}}{(2n+1)!}}} {\displaystyle {\frac {2^{n+1}(n!)^{2}}{(2n+1)!}}}
e a x {\displaystyle e^{ax},円} {\displaystyle e^{ax},円} 2 π a I n + 1 / 2 ( a ) {\displaystyle {\sqrt {\frac {2\pi }{a}}}I_{n+1/2}(a)} {\displaystyle {\sqrt {\frac {2\pi }{a}}}I_{n+1/2}(a)}
e i a x {\displaystyle e^{iax},円} {\displaystyle e^{iax},円} 2 π a i n J n + 1 / 2 ( a ) {\displaystyle {\sqrt {\frac {2\pi }{a}}}i^{n}J_{n+1/2}(a)} {\displaystyle {\sqrt {\frac {2\pi }{a}}}i^{n}J_{n+1/2}(a)}
x f ( x ) {\displaystyle xf(x),円} {\displaystyle xf(x),円} 1 2 n + 1 [ ( n + 1 ) f ~ ( n + 1 ) + n f ~ ( n − 1 ) ] {\displaystyle {\frac {1}{2n+1}}[(n+1){\tilde {f}}(n+1)+n{\tilde {f}}(n-1)]} {\displaystyle {\frac {1}{2n+1}}[(n+1){\tilde {f}}(n+1)+n{\tilde {f}}(n-1)]}
( 1 − x 2 ) − 1 / 2 {\displaystyle (1-x^{2})^{-1/2},円} {\displaystyle (1-x^{2})^{-1/2},円} π P n 2 ( 0 ) {\displaystyle \pi P_{n}^{2}(0)} {\displaystyle \pi P_{n}^{2}(0)}
[ 2 ( a − x ) ] − 1 {\displaystyle [2(a-x)]^{-1},円} {\displaystyle [2(a-x)]^{-1},円} Q n ( a ) {\displaystyle Q_{n}(a)} {\displaystyle Q_{n}(a)}
( 1 − 2 a x + a 2 ) − 1 / 2 ,   | a | < 1 {\displaystyle (1-2ax+a^{2})^{-1/2},\ |a|<1,円} {\displaystyle (1-2ax+a^{2})^{-1/2},\ |a|<1,円} 2 a n ( 2 n + 1 ) − 1 {\displaystyle 2a^{n}(2n+1)^{-1}} {\displaystyle 2a^{n}(2n+1)^{-1}}
( 1 − 2 a x + a 2 ) − 3 / 2 ,   | a | < 1 {\displaystyle (1-2ax+a^{2})^{-3/2},\ |a|<1,円} {\displaystyle (1-2ax+a^{2})^{-3/2},\ |a|<1,円} 2 a n ( 1 − a 2 ) − 1 {\displaystyle 2a^{n}(1-a^{2})^{-1}} {\displaystyle 2a^{n}(1-a^{2})^{-1}}
∫ 0 a t b − 1 d t ( 1 − 2 x t + t 2 ) 1 / 2 ,   | a | < 1   b > 0 {\displaystyle \int _{0}^{a}{\frac {t^{b-1},円dt}{(1-2xt+t^{2})^{1/2}}},\ |a|<1\ b>0,円} {\displaystyle \int _{0}^{a}{\frac {t^{b-1},円dt}{(1-2xt+t^{2})^{1/2}}},\ |a|<1\ b>0,円} 2 a n + b ( 2 n + 1 ) ( n + b ) {\displaystyle {\frac {2a^{n+b}}{(2n+1)(n+b)}}} {\displaystyle {\frac {2a^{n+b}}{(2n+1)(n+b)}}}
d d x [ ( 1 − x 2 ) d d x ] f ( x ) {\displaystyle {\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x),円} {\displaystyle {\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x),円} − n ( n + 1 ) f ~ ( n ) {\displaystyle -n(n+1){\tilde {f}}(n)} {\displaystyle -n(n+1){\tilde {f}}(n)}
{ d d x [ ( 1 − x 2 ) d d x ] } k f ( x ) {\displaystyle \left\{{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]\right\}^{k}f(x),円} {\displaystyle \left\{{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]\right\}^{k}f(x),円} ( − 1 ) k n k ( n + 1 ) k f ~ ( n ) {\displaystyle (-1)^{k}n^{k}(n+1)^{k}{\tilde {f}}(n)} {\displaystyle (-1)^{k}n^{k}(n+1)^{k}{\tilde {f}}(n)}
f ( x ) 4 − d d x [ ( 1 − x 2 ) d d x ] f ( x ) {\displaystyle {\frac {f(x)}{4}}-{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x),円} {\displaystyle {\frac {f(x)}{4}}-{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x),円} ( n + 1 2 ) 2 f ~ ( n ) {\displaystyle \left(n+{\frac {1}{2}}\right)^{2}{\tilde {f}}(n)} {\displaystyle \left(n+{\frac {1}{2}}\right)^{2}{\tilde {f}}(n)}
ln ⁡ ( 1 − x ) {\displaystyle \ln(1-x),円} {\displaystyle \ln(1-x),円} { 2 ( ln ⁡ 2 − 1 ) , n = 0 − 2 n ( n + 1 ) , n > 0 {\displaystyle {\begin{cases}2(\ln 2-1),&n=0\\-{\frac {2}{n(n+1)}},&n>0\end{cases}},円} {\displaystyle {\begin{cases}2(\ln 2-1),&n=0\\-{\frac {2}{n(n+1)}},&n>0\end{cases}},円}
f ( x ) ∗ g ( x ) {\displaystyle f(x)*g(x),円} {\displaystyle f(x)*g(x),円} f ~ ( n ) g ~ ( n ) {\displaystyle {\tilde {f}}(n){\tilde {g}}(n)} {\displaystyle {\tilde {f}}(n){\tilde {g}}(n)}
∫ − 1 x f ( t ) d t {\displaystyle \int _{-1}^{x}f(t),円dt,円} {\displaystyle \int _{-1}^{x}f(t),円dt,円} { f ~ ( 0 ) − f ~ ( 1 ) , n = 0 f ~ ( n − 1 ) − f ~ ( n + 1 ) 2 n + 1 , n > 1 {\displaystyle {\begin{cases}{\tilde {f}}(0)-{\tilde {f}}(1),&n=0\\{\frac {{\tilde {f}}(n-1)-{\tilde {f}}(n+1)}{2n+1}},&n>1\end{cases}},円} {\displaystyle {\begin{cases}{\tilde {f}}(0)-{\tilde {f}}(1),&n=0\\{\frac {{\tilde {f}}(n-1)-{\tilde {f}}(n+1)}{2n+1}},&n>1\end{cases}},円}
d d x g ( x ) ,   g ( x ) = ∫ − 1 x f ( t ) d t {\displaystyle {\frac {d}{dx}}g(x),\ g(x)=\int _{-1}^{x}f(t),円dt} {\displaystyle {\frac {d}{dx}}g(x),\ g(x)=\int _{-1}^{x}f(t),円dt} g ( 1 ) − ∫ − 1 1 g ( x ) d d x P n ( x ) d x {\displaystyle g(1)-\int _{-1}^{1}g(x){\frac {d}{dx}}P_{n}(x),円dx} {\displaystyle g(1)-\int _{-1}^{1}g(x){\frac {d}{dx}}P_{n}(x),円dx}

References

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  1. ^ Debnath, Lokenath; Dambaru Bhatta (2007). Integral transforms and their applications (2nd ed.). Boca Raton: Chapman & Hall/CRC. ISBN 9781482223576.
  2. ^ Churchill, R. V. (1954). "The Operational Calculus of Legendre Transforms". Journal of Mathematics and Physics. 33 (1–4): 165–178. doi:10.1002/sapm1954331165. hdl:2027.42/113680 .
  3. ^ Churchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100.


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