积分变换

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）。

積分變換	符號	核 ${\displaystyle K}$	f(t)	t1	t2	反核 ${\displaystyle K^{-1}}$	u1	u2
傅立葉變換	${\displaystyle {\mathcal {F}}}$	${\displaystyle {\frac {e^{-iut}}{\sqrt {2\pi }}}}$	${\displaystyle L_{1}}$	${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {e^{+iut}}{\sqrt {2\pi }}}}$	${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$
傅立葉正弦變換	${\displaystyle {\mathcal {F}}_{s}}$	${\displaystyle {\frac {{\sqrt {2}}\sin {(ut)}}{\sqrt {\pi }}}}$	on ${\displaystyle [0,\infty )}$, real-valued	${\displaystyle 0\,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {{\sqrt {2}}\sin {(ut)}}{\sqrt {\pi }}}}$	${\displaystyle 0\,}$	${\displaystyle \infty \,}$
傅立葉餘弦變換	${\displaystyle {\mathcal {F}}_{c}}$	${\displaystyle {\frac {{\sqrt {2}}\cos {(ut)}}{\sqrt {\pi }}}}$	on ${\displaystyle [0,\infty )}$, real-valued	${\displaystyle 0\,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {{\sqrt {2}}\cos {(ut)}}{\sqrt {\pi }}}}$	${\displaystyle 0\,}$	${\displaystyle \infty \,}$
Hartley变换	${\displaystyle {\mathcal {H}}}$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$		${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$	${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$
Mellin变换	${\displaystyle {\mathcal {M}}}$	${\displaystyle t^{u-1}\,}$		${\displaystyle 0\,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {t^{-u}}{2\pi i}}\,}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
双边拉普拉斯变换	${\displaystyle {\mathcal {B}}}$	${\displaystyle e^{-ut}\,}$		${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {e^{+ut}}{2\pi i}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
拉普拉斯变换	${\displaystyle {\mathcal {L}}}$	${\displaystyle e^{-ut}\,}$		${\displaystyle 0\,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {e^{+ut}}{2\pi i}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
魏尔斯特拉斯变换	${\displaystyle {\mathcal {W}}}$	${\displaystyle {\frac {e^{-(u-t)^{2}/4}}{\sqrt {4\pi }}}\,}$		${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {e^{+(u-t)^{2}/4}}{i{\sqrt {4\pi }}}}}$	${\displaystyle c\!-\!i\infty }$	${\displaystyle c\!+\!i\infty }$
Hankel变换		${\displaystyle t\,J_{\nu }(ut)}$		${\displaystyle 0\,}$	${\displaystyle \infty \,}$	${\displaystyle u\,J_{\nu }(ut)}$	${\displaystyle 0\,}$	${\displaystyle \infty \,}$
阿贝尔积分变换		${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}}$		${\displaystyle u\,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}}$	${\displaystyle t\,}$	${\displaystyle \infty \,}$
希爾伯特轉換	${\displaystyle {\mathcal {H}}il}$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$		${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$	${\displaystyle -\infty \,}$	${\displaystyle \infty \,}$
泊松核		${\displaystyle {\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}}$		${\displaystyle 0\,}$	${\displaystyle 2\pi \,}$
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。

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