积分变换

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積分變換（integral transform）是數學中作用于函数的算子，用以處理微分方程等問題。常見的有傅里葉變換、拉普拉斯變換等。

以一變數為 ${\displaystyle t}$ 的函數 ${\displaystyle f(t)}$ 為例，${\displaystyle f(t)}$ 經過一積分轉換 ${\displaystyle T}$ 得到 ${\displaystyle Tf(u)}$：

${\displaystyle (Tf)(u)=\int \limits _{t_{1}}^{t_{2}}K(t,u)\,f(t)\,dt}$

其中 ${\displaystyle K}$ 是个确定的二元函数, 稱為此積分變換的核函數（kernel function）或核（nucleus）。当选取不同的积分域和变换核时，就得到不同名称的积分变换。${\displaystyle f(t)}$ 称为象原函数，${\displaystyle Tf(u)}$ 称为 ${\displaystyle f(t)}$ 的象函数，在一定条件下，它们是一一对应而变换是可逆的。

有些積分變換有相對應的反積分變換（inverse transform），使得

${\displaystyle f(t)=\int \limits _{u_{1}}^{u_{2}}K^{-1}(u,t)\,(Tf)(u)\,du}$

而 ${\displaystyle K^{-1}(u,t)}$ 稱為反核（inverse kernel）。

积分变换	符号	核K	f(t)	t1	t2	反核K−1	u1	u2
阿贝尔积分变换（英语：Abel transform）	F, f	${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}}$		u	${\displaystyle \infty }$	${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}}$ [1]	t	${\displaystyle \infty }$
相关 Legendre 变换（Associated Legendre transform）	${\displaystyle {\mathcal {J}}_{n,m}}$	${\displaystyle (1-x^{2})^{-m/2}P_{n}^{m}(x)}$		${\displaystyle -1}$	${\displaystyle 1}$		${\displaystyle 0}$	${\displaystyle \infty }$
傅里叶变换	${\displaystyle {\mathcal {F}}}$	${\displaystyle e^{-2\pi iut}}$	${\displaystyle L_{1}}$	${\displaystyle -\infty }$	${\displaystyle \infty }$	${\displaystyle e^{2\pi iut}}$	${\displaystyle -\infty }$	${\displaystyle \infty }$
傅里叶正弦变换	${\displaystyle {\mathcal {F}}_{s}}$	${\displaystyle {\sqrt {\frac {2}{\pi }}}\sin(ut)}$	on ${\displaystyle [0,\infty )}$, real-valued	${\displaystyle 0}$	${\displaystyle \infty }$	${\displaystyle {\sqrt {\frac {2}{\pi }}}\sin(ut)}$	${\displaystyle 0}$	${\displaystyle \infty }$
傅里叶余弦变换	${\displaystyle {\mathcal {F}}_{c}}$	${\displaystyle {\sqrt {\frac {2}{\pi }}}\cos(ut)}$	on ${\displaystyle [0,\infty )}$, real-valued	0	${\displaystyle \infty }$	${\displaystyle {\sqrt {\frac {2}{\pi }}}\cos(ut)}$	0	${\displaystyle \infty }$
汉克尔变换		${\displaystyle t\,J_{\nu }(ut)}$		0	${\displaystyle \infty }$	${\displaystyle u\,J_{\nu }(ut)}$	0	${\displaystyle \infty }$
Hartley变换（英语：Hartley transform）	${\displaystyle {\mathcal {H}}}$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$		${\displaystyle -\infty }$	${\displaystyle \infty }$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$	${\displaystyle -\infty }$	${\displaystyle \infty }$
Hermite变换（英语：Hermite_transform）	${\displaystyle H}$	${\displaystyle e^{-x^{2}}H_{n}(x)}$		${\displaystyle -\infty }$	${\displaystyle \infty }$		${\displaystyle 0}$	${\displaystyle \infty }$
希尔伯特变换	${\displaystyle {\mathcal {H}}il}$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$		${\displaystyle -\infty }$	${\displaystyle \infty }$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$	${\displaystyle -\infty }$	${\displaystyle \infty }$
Jacobi变换（英语：Jacobi_transform）	${\displaystyle J}$	${\displaystyle (1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)}$		${\displaystyle -1}$	${\displaystyle 1}$		${\displaystyle 0}$	${\displaystyle \infty }$
Laguerre变换（英语：Laguerre_transform）	${\displaystyle L}$	${\displaystyle e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)}$		${\displaystyle 0}$	${\displaystyle \infty }$		${\displaystyle 0}$	${\displaystyle \infty }$
拉普拉斯变换	${\displaystyle {\mathcal {L}}}$	e−ut		0	${\displaystyle \infty }$	${\displaystyle {\frac {e^{ut}}{2\pi i}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
Legendre变换（英语：Legendre_transform_(integral_transform)）	${\displaystyle {\mathcal {J}}}$	${\displaystyle P_{n}(x)\,}$		${\displaystyle -1}$	${\displaystyle 1}$		${\displaystyle 0}$	${\displaystyle \infty }$
梅林变换	${\displaystyle {\mathcal {M}}}$	tu−1		0	${\displaystyle \infty }$	${\displaystyle {\frac {t^{-u}}{2\pi i}}\,}$[2]	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
双边拉普拉斯变换	${\displaystyle {\mathcal {B}}}$	e−ut		${\displaystyle -\infty }$	${\displaystyle \infty }$	${\displaystyle {\frac {e^{ut}}{2\pi i}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
泊松核（英语：Poisson kernel）		${\displaystyle {\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}}$		0	2π
拉东变换	Rƒ			${\displaystyle -\infty }$	${\displaystyle \infty }$
魏尔斯特拉斯变换（英语：Weierstrass transform）	${\displaystyle {\mathcal {W}}}$	${\displaystyle {\frac {e^{-{\frac {(u-t)^{2}}{4}}}}{\sqrt {4\pi }}}\,}$		${\displaystyle -\infty }$	${\displaystyle \infty }$	${\displaystyle {\frac {e^{\frac {(u-t)^{2}}{4}}}{i{\sqrt {4\pi }}}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
X-ray变换（英语：X-ray_transform）	Xƒ			${\displaystyle -\infty }$	${\displaystyle \infty }$
狄拉克δ函数		${\displaystyle \delta (u-t)\,}$		${\displaystyle t_{1}<u\,}$	${\displaystyle t_{2}>u\,}$	${\displaystyle \delta (t-u)\,}$	${\displaystyle u_{1}\!<\!t}$	${\displaystyle u_{2}\!>\!t}$

在反積分轉換中, 常數c 由積分函數決定。

• 线性微分方程
• 拉普拉斯变换
• 常微分方程
• 偏微分方程
• 初值問題
• 边值问题

1. ^ Assuming the Abel transform is not discontinuous at ${\displaystyle u}$.
2. ^ Some conditions apply, see Mellin inversion theorem for details.