# 积分变换

## 概述

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

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

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

## 積分變換表列

en:Hartley transform ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$ ${\displaystyle -\infty \,}$ ${\displaystyle \infty \,}$ ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$ ${\displaystyle -\infty \,}$ ${\displaystyle \infty \,}$
en:Mellin transform ${\displaystyle {\mathcal {M}}}$ ${\displaystyle t^{u-1}\,}$ ${\displaystyle 0\,}$ ${\displaystyle \infty \,}$ ${\displaystyle {\frac {t^{-u}}{2\pi i}}\,}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$
Two-sided Laplace
transform
${\displaystyle {\mathcal {B}}}$ ${\displaystyle e^{-ut}\,}$ ${\displaystyle -\infty \,}$ ${\displaystyle \infty \,}$ ${\displaystyle {\frac {e^{+ut}}{2\pi i}}}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$

en:Weierstrass transform ${\displaystyle {\mathcal {W}}}$ ${\displaystyle {\frac {e^{-(u-t)^{2}/4}}{\sqrt {4\pi }}}\,}$ ${\displaystyle -\infty \,}$ ${\displaystyle \infty \,}$ ${\displaystyle {\frac {e^{+(u-t)^{2}/4}}{i{\sqrt {4\pi }}}}}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$
en:Hankel transform ${\displaystyle t\,J_{\nu }(ut)}$ ${\displaystyle 0\,}$ ${\displaystyle \infty \,}$ ${\displaystyle u\,J_{\nu }(ut)}$ ${\displaystyle 0\,}$ ${\displaystyle \infty \,}$
en:Abel transform ${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}}$ ${\displaystyle u\,}$ ${\displaystyle \infty \,}$ ${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}}$ ${\displaystyle t\,}$ ${\displaystyle \infty \,}$

en:Poisson kernel ${\displaystyle {\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}}$ ${\displaystyle 0\,}$ ${\displaystyle 2\pi \,}$
Identity transform ${\displaystyle \delta (u-t)\,}$ ${\displaystyle t_{1} ${\displaystyle t_{2}>u\,}$ ${\displaystyle \delta (t-u)\,}$ ${\displaystyle u_{1}\!<\!t}$ ${\displaystyle u_{2}\!>\!t}$