User:Physikerwelt/sandbox/FT

$f(t)$

${\hat {f}}(\omega )$

$\omega$

$g(t)$

$\scriptstyle f(t)$

$\scriptstyle {\hat {f}}(\omega )$

$\scriptstyle g(t)$

$\scriptstyle {\hat {g}}(\omega )$

$\scriptstyle t$

$\scriptstyle \omega$

$\scriptstyle t$

$\scriptstyle \omega$

${\hat {f}}$

$f:\mathbb {R} \rightarrow \mathbb {C}$

${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,$

$f$

${\hat {f}}$

$f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi i\xi x}\,d\xi ,$

$f$

${\hat {f}}$

$f$

${\hat {f}}$

$L^{2}$

$L^{1}$

$L^{\infty }$

${\hat {f}}$

$c_{n}={\frac {1}{T}}\int _{-T/2}^{T/2}f(x)\ e^{-2\pi i(n/T)x}\,dx.$

$c_{n}=(1/T){\hat {f}}(n/T)$

$f(x)=\sum _{n=-\infty }^{\infty }c_{n}\ e^{2\pi i(n/T)x}=\sum _{n=-\infty }^{\infty }{\hat {f}}(\xi _{n})\ e^{2\pi i\xi _{n}x}\Delta \xi ,$

${\hat {f}}(3)$

${\hat {f}}(5)$

$\int _{-\infty }^{\infty }|f(x)|\,dx<\infty .$

${\hat {f}}(\xi )$

${\hat {g}}(\xi )$

${\hat {h}}(\xi )$

${\hat {h}}(\xi )=a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi ).$

$h(x)=f(x-x_{0}),$

${\hat {h}}(\xi )=e^{-i\,2\pi \,x_{0}\,\xi }{\hat {f}}(\xi ).$

$h(x)=e^{i\,2\pi \,x\,\xi _{0}}f(x),$

${\hat {h}}(\xi )={\hat {f}}(\xi -\xi _{0}).$

$h(x)=f(ax)$

${\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right).$

${\hat {h}}(\xi )={\hat {f}}(-\xi ).$

$h(x)={\overline {f(x)}},$

${\hat {h}}(\xi )={\overline {{\hat {f}}(-\xi )}}.$

${\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}$

${\hat {f}}$

${\hat {f}}(-\xi )=-{\overline {{\hat {f}}(\xi )}}.$

$\xi =0$

${\hat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.$

$\xi =0$

${\hat {f}}.$

${\mathcal {F}},$

${\mathcal {F}}(f):={\hat {f}},$

${\mathcal {F}}^{2}(f)(x)=f(-x),$

${\mathcal {F}}^{4}(f)=f,$

${\mathcal {F}}^{3}({\hat {f}})=f.$

${\mathcal {P}}$

${\mathcal {P}}[f]\colon t\mapsto f(-t),$

${\mathcal {F}}^{0}=\mathrm {Id} ,\qquad {\mathcal {F}}^{1}={\mathcal {F}},\qquad {\mathcal {F}}^{2}={\mathcal {P}},\qquad {\mathcal {F}}^{4}=\mathrm {Id}$

${\mathcal {F}}^{3}={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}}$

$t$

$\xi$

$L^{2}(bR)$(was L^2(\b R))

$\xi$

$t$

$t$

$\xi$

$t$

$t$

$2\pi$

$\xi$

$t$

$\xi$

$t$

$t$

$\xi$

$\xi$

$t$

$t$

$\xi$

$\omega =2\pi \xi$

${\hat {x}}_{1}$

${\hat {x}}$

${\hat {x}}_{1}(\omega )={\hat {x}}\left({\omega \over 2\pi }\right)=\int _{-\infty }^{\infty }x(t)e^{-i\omega t}\,dt$

$x(t)={1 \over {2\pi }}\int _{-\infty }^{\infty }{\hat {x}}_{1}(\omega )e^{it\omega }\,d\omega .$

${\sqrt {2\pi }}$

${\hat {x}}_{2}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x(t)e^{-i\omega t}\,dt,$

$x(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {x}}_{2}(\omega )e^{it\omega }\,d\omega .$

$i$

$i$

$i$

$-i$

$\phi$

$f$

$X$

$x$

$2\pi$

$\phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.$

${\hat {f}}$

$\|{\hat {f}}\|_{\infty }\leq \|f\|_{1}$

${\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .$

${\hat {f}}$

${\hat {f}}$

$f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{2i\pi x\xi }\,d\xi$

${\hat {f}}(\xi )$

${\hat {g}}(\xi )$

$\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,{\rm {d}}x=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,$

$\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .$

$f$

$\sum _{n}{\hat {f}}(n)=\sum _{n}f(n).$

${\widehat {f'\;}}(\xi )=2\pi i\xi {\hat {f}}(\xi ).$

${\widehat {f^{(n)}}}(\xi )=(2\pi i\xi )^{n}{\hat {f}}(\xi ).$

$f(x)$

${\hat {f}}(\xi )$

$|\xi |\to \infty$

$f(x)$

$|x|\to \infty$

${\hat {f}}(\xi )$

${\hat {f}}(\xi )$

${\hat {g}}(\xi )$

${\hat {f}}(\xi )$

${\hat {g}}(\xi )$

$h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,$

${\hat {h}}(\xi )={\hat {f}}(\xi )\cdot {\hat {g}}(\xi ).$

${\hat {g}}(\xi )$

${\hat {p}}(\xi )$

${\hat {q}}(\xi )$

$h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}\,g(x+y)\,dy$

${\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,\cdot \,{\hat {g}}(\xi ).$

$h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy$

${\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {f}}(\xi )=|{\hat {f}}(\xi )|^{2}.$

${\psi }_{n}(x)={\frac {2^{1/4}}{\sqrt {n!}}}\,e^{-\pi x^{2}}\mathrm {He} _{n}(2x{\sqrt {\pi }}),$

$\mathrm {He} _{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {x^{2}}{2}}}$

${\hat {\psi }}_{n}(\xi )=(-i)^{n}{\psi }_{n}(\xi )$

${\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-2\pi i\xi t}f(t)\,dt$

$\xi$

$f$

$\xi =\sigma +i\tau$

$f$

$n$

$n$

${\hat {f}}(\sigma +i\tau )$

$a>0$

$n\geq 0$

$\vert \xi ^{n}{\hat {f}}(\xi )\vert \leq Ce^{a\vert \tau \vert }$

$C$

$f$

$[-a,a]$

${\hat {f}}$

$\sigma$

$\tau$

$\tau$

$\sigma$

$f$

$L^{2}$

$n=0$

$f$

$t\geq 0$

$f$

${\hat {f}}$

$\tau <0$

$\tau$

${\hat {f}}(\xi )$

$F(s)$

$s$

$f$

$f(t)$

$\vert f(t)\vert <Ce^{a\vert t\vert }$

$C,a\geq 0$

${\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,$

$2\pi \tau <-a$

$f$

$F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.$

$f$

${\hat {f}}(i\tau )=F(-2\pi \tau ).$

$s=2\pi i\xi$

${\hat {f}}$

$a\leq \tau \leq b$

$\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{2\pi i\xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{2\pi i\xi t}\,d\sigma$

$f(t)=0$

$t<0$

$\vert f(t)\vert <Ce^{a\vert t\vert }$

$C,a>0$

$f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{2\pi i\xi t}\,d\sigma ,$

$\tau <-{a \over 2\pi }$

$f(t)={\frac {1}{2\pi i}}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}ds$

$b>a$

$F(s)$

$f(t)$

$f(t)e^{-at}$

$L^{1}$

$t$

$f$

${\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-2\pi i\mathbf {x} \cdot {\boldsymbol {\xi }}}\,d\mathbf {x}$

$\left\langle \mathbf {x} ,{\boldsymbol {\xi }}\right\rangle$

${\hat {f}}(\xi )$

$\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.$

${\hat {f}}(\xi )$

$D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.$

$D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}$

$f(x)=C_{1}\,e^{-\pi x^{2}/\sigma ^{2}}$

${\hat {f}}(\xi )=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}$

$C_{1}={\sqrt[{4}]{2}}/{\sqrt {\sigma }}$

$\left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}$

$H(|f|^{2})+H(|{\hat {f}}|^{2})\geq \log(e/2)$

$H(p)=-\int _{-\infty }^{\infty }p(x)\log(p(x))\,dx$

$f$

$\lambda$

$a$

$b$

$a$

$b$

${\hat {f}}(\xi )=i^{-k}f(\xi )$

${\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )$

$F_{0}(r)=2\pi i^{-k}r^{-(n+2k-2)/2}\int _{0}^{\infty }f_{0}(s)J_{(n+2k-2)/2}(2\pi rs)s^{(n+2k)/2}\,ds.$

$f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{2\pi ix\cdot \xi }\,d\xi ,\quad x\in \mathbf {R} ^{n}.$

${\hat {f}}(\xi )=\int _{\mathbf {R} ^{n}}f(x)e^{-2\pi i\xi \cdot x}\,dx$

${\mathcal {F}}$

${\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-2\pi ix\cdot \xi }\,dx$

${\mathcal {F}}$

$\int _{\mathbf {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbf {R} ^{n}}f(x){\hat {g}}(x)\,dx.$

$T_{f}(\varphi )=\int _{\mathbf {R} ^{n}}f(x)\varphi (x)\,dx$

${\hat {T}}_{f}$

${\hat {T}}_{f}(\varphi )=T_{f}({\hat {\varphi }})$

${\hat {\mu }}(\xi )=\int _{\mathbf {R} ^{n}}\mathrm {e} ^{-2\pi ix\cdot \xi }\,d\mu .$

${\hat {G}}$

${\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \qquad {\text{for any }}\xi \in {\hat {G}}.$

${\hat {f}}(\xi )$

${\hat {G}}$

$f^{*}(g)={\overline {f(g^{-1})}}.$

$a\mapsto (\varphi \mapsto \varphi (a))$

$\langle {\hat {\mu }}\xi ,\eta \rangle _{H_{\sigma }}=\int _{G}\langle {\overline {U}}_{g}^{(\sigma )}\xi ,\eta \rangle \,d\mu (g)$

${\overline {U}}^{(\sigma )}$

$d\mu =f\,d\lambda$

$\mu \mapsto {\hat {\mu }}$

$\|E\|=\sup _{\sigma \in \Sigma }\|E_{\sigma }\|$

$f^{*}(g)={\overline {f(g^{-1})}},$

$f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} ({\hat {f}}(\sigma )U_{g}^{(\sigma )})$

$y(x,0)=f(x),{\partial y(x,t) \over \partial t}=g(x).$

$f$

$g$

$y$

${\hat {y}}$

${\hat {y}}$

$y$

$\cos \left(2\pi \xi (x\pm t)\right){\mbox{ or }}\sin \left(2\pi \xi (x\pm t)\right)$

$y(x,t)=\int _{0}^{\infty }a_{+}(\xi )\cos \left(2\pi \xi (x+t)\right)+a_{-}(\xi )\cos \left(2\pi \xi (x-t)\right)+b_{+}(\xi )\sin \left(2\pi \xi (x+t)\right)+b_{-}(\xi )\sin \left(2\pi \xi (x-t)\right)\,d\xi$

$a_{+}$

$a_{-}$

$b_{+}$

$b_{-}$

$a_{\pm }$

$b_{\pm }$

$x$

$a_{\pm }$

$b_{\pm }$

$y$

$t=0$

$t=0$

$x$

$2\int _{-\infty }^{\infty }y(x,0)\cos \left(2\pi \xi x\right)\,dx=a_{+}+a_{-}$

$2\int _{-\infty }^{\infty }y(x,0)\sin \left(2\pi \xi x\right)\,dx=b_{+}+b_{-}.$

$y$

$t$

$2\int _{-\infty }^{\infty }{\partial y(u,0) \over \partial t}\sin(2\pi \xi x)\,dx=(2\pi \xi )(-a_{+}+a_{-})$

$2\int _{-\infty }^{\infty }{\partial y(u,0) \over \partial t}\cos(2\pi \xi x)\,dx=(2\pi \xi )(b_{+}-b_{-}).$

$a_{\pm }$

$b_{\pm }$

$\xi$

$\xi$

$f$

$g$

$a_{\pm }$

$b_{\pm }$

$f$

$g$

$x$

$t$

${\hat {y}}$

$y(x,t)$

$L^{1}$

$x$

$2\pi i\xi$

$t$

$2\pi if$

$f$

${\hat {y}}$

$\xi ^{2}{\hat {y}}(\xi ,f)=f^{2}{\hat {y}}(\xi ,f).$

${\hat {y}}(\xi ,f)=0$

$\xi =\pm f$

${\hat {f}}=\delta (\xi \pm f)$

$\xi ^{2}-f^{2}=0$

$\xi =f$

$\xi =-f$

$\phi$

$\int \int {\hat {y}}\phi (\xi ,f)\,d\xi \,df=\int s_{+}\phi (\xi ,\xi )\,d\xi +\int s_{-}\phi (\xi ,-\xi )\,d\xi ,$

$s_{+}$

$s_{-}$

$\phi (\xi ,f)=e^{2\pi i(x\xi +tf)}$

$y(x,0)=\int \{s_{+}(\xi )+s_{-}(\xi )\}e^{2\pi i\xi x+0}\,d\xi$

${\partial y(x,0) \over \partial t}=$

$x$

$x$

$s_{\pm }$

$L^{1}$

$L^{2}$

$q$

$p$

$q$

$p$

$p$

$q$

$p$

$q$

$p$

$q$

$q$

$p$

$L^{2}$

$L^{2}$

$q$

$p$

$i$

$V(x)$

$\psi$

$t=0$

$R$

$f$

$\tau$

$f$

$f$

$R$

$\tau$

$\tau =$

$f$

$f$

$f(t)$

$t$

$f$

$f$

$P$

$\xi$

${\hat {f}}(\xi )$

${\tilde {f}}(\xi ),\ {\tilde {f}}(\omega ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}(\omega ),\ F(\omega ),\ {\mathcal {F}}(j\omega ),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}\left(f(t)\right),\ {\mathcal {F}}\{f(t)\}.$

${\hat {f}}(\xi )$

${\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}$

$A(\xi )=|{\hat {f}}(\xi )|,$

$\varphi (\xi )=\arg {\big (}{\hat {f}}(\xi ){\big )},$

$f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i(2\pi \xi x+\varphi (\xi ))}\,d\xi ,$

${\mathcal {F}}$

${\mathcal {F}}(f)$

${\mathcal {F}}$

${\mathcal {F}}f$

${\mathcal {F}}(f)$

${\mathcal {F}}f(\xi )$

$({\mathcal {F}}f)(\xi )$

${\mathcal {F}}$

${\mathcal {F}}(f(x))$

${\mathcal {F}}(\mathrm {rect} (x))=\mathrm {sinc} (\xi )$

${\mathcal {F}}(f(x+x_{0}))={\mathcal {F}}(f(x))e^{2\pi i\xi x_{0}}$

$\omega =2\pi \xi ,$

${\hat {f}}(\omega )=\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx.$

$f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbf {R} ^{n}}{\hat {f}}(\omega )e^{i\omega \cdot x}\,d\omega .$

${\hat {f}}(\omega )={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx,$

$f(x)={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}{\hat {f}}(\omega )e^{i\omega \cdot x}\,d\omega .$

$\displaystyle {\hat {f}}_{1}(\xi )\ {\stackrel {\mathrm {def} }{=}}\ \int _{\mathbf {R} ^{n}}f(x)e^{-2\pi ix\cdot \xi }\,dx={\hat {f}}_{2}(2\pi \xi )=(2\pi )^{n/2}{\hat {f}}_{3}(2\pi \xi )$

$\displaystyle {\hat {f}}_{3}(\omega ){\stackrel {\mathrm {def} }{{}={}}}{\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{1}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{2}(\omega )$

$\displaystyle {\hat {f}}_{2}(\omega )\ {\stackrel {\mathrm {def} }{=}}\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\hat {f}}_{1}\!\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{n/2}{\hat {f}}_{3}(\omega )$

$E(e^{it\cdot X})=\int e^{it\cdot x}\,d\mu _{X}(x)$

${\hat {f}}$

${\hat {g}}$

${\hat {h}}$

$\displaystyle f(x)\,$

$\displaystyle {\hat {f}}(\xi )=$

$\displaystyle {\hat {f}}(\omega )=$

$\displaystyle {\hat {f}}(\nu )=$

$\displaystyle a\cdot f(x)+b\cdot g(x)\,$

$\displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,$

$\displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,$

$\displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,$

$\displaystyle f(x-a)\,$

$\displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,$

$\displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,$

$\displaystyle e^{-ia\nu }{\hat {f}}(\nu )\,$

$\displaystyle e^{2\pi iax}f(x)\,$

$\displaystyle {\hat {f}}(\xi -a)\,$

$\displaystyle {\hat {f}}(\omega -2\pi a)\,$

$\displaystyle {\hat {f}}(\nu -2\pi a)\,$

$\displaystyle f(ax)\,$

$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,$

$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,$

$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,$

$\displaystyle |a|\,$

$\displaystyle f(ax)\,$

$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,$

$\displaystyle {\hat {f}}(x)\,$

$\displaystyle f(-\xi )\,$

$\displaystyle f(-\omega )\,$

$\displaystyle 2\pi f(-\nu )\,$

${\hat {f}}$

$x$

$\xi$

$\omega$

$\nu$

$\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,$

$\displaystyle (2\pi i\xi )^{n}{\hat {f}}(\xi )\,$

$\displaystyle (i\omega )^{n}{\hat {f}}(\omega )\,$

$\displaystyle (i\nu )^{n}{\hat {f}}(\nu )\,$

$\displaystyle x^{n}f(x)\,$

$\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,$

$\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}$

$\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\nu )}{d\nu ^{n}}}$

$\displaystyle (f*g)(x)\,$

$\displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,$

$\displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,$

$\displaystyle {\hat {f}}(\nu ){\hat {g}}(\nu )\,$

$\displaystyle f*g\,$

$f$

$g$

$\displaystyle f(x)g(x)\,$

$\displaystyle ({\hat {f}}*{\hat {g}})(\xi )\,$

$\displaystyle ({\hat {f}}*{\hat {g}})(\omega ) \over {\sqrt {2\pi }}\,$

$\displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\nu )\,$

$\displaystyle f(x)\,$

$\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,$

$\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,$

$\displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,$

$\displaystyle {\overline {z}}\,$

$\displaystyle f(x)\,$

$\displaystyle {\hat {f}}(\omega )$

$\displaystyle {\hat {f}}(\xi )$

$\displaystyle {\hat {f}}(\nu )\,$

$\displaystyle f(x)\,$

$\displaystyle {\hat {f}}(\omega )$

$\displaystyle {\hat {f}}(\xi )$

$\displaystyle {\hat {f}}(\nu )$

$\displaystyle {\overline {f(x)}}$

$\displaystyle {\overline {{\hat {f}}(-\xi )}}$

$\displaystyle {\overline {{\hat {f}}(-\omega )}}$

$\displaystyle {\overline {{\hat {f}}(-\nu )}}$

$\displaystyle f(x)$

$\displaystyle {\hat {f}}(\xi )=$

$\displaystyle {\hat {f}}(\omega )=$

$\displaystyle {\hat {f}}(\nu )=$

$\displaystyle \operatorname {rect} (ax)\,$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right)$

$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right)$

$\displaystyle \operatorname {sinc} (ax)\,$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\,$

$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right)$

$\displaystyle \operatorname {sinc} ^{2}(ax)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right)$

$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right)$

$\displaystyle \operatorname {tri} (ax)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,$

$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)$

$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right)$

$\displaystyle e^{-ax}u(x)\,$

$\displaystyle {\frac {1}{a+2\pi i\xi }}$

$\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}$

$\displaystyle {\frac {1}{a+i\nu }}$

$\displaystyle e^{-\alpha x^{2}}\,$

$\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}$

$\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$

$\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha }}}$

$\displaystyle \operatorname {e} ^{-a|x|}\,$

$\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}$

$\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}$

$\displaystyle {\frac {2a}{a^{2}+\nu ^{2}}}$

$\displaystyle \operatorname {sech} (ax)\,$

$\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)$

$\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)$

$\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right)$

$\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,$

$\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}$

$\cdot e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)$

$\displaystyle {\frac {(-i)^{n}}{a}}$

$\cdot e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)$

$\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}$

$\cdot e^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{a}}\right)$

$H_{n}$

$\displaystyle f(x)$

$\displaystyle {\hat {f}}(\xi )=$

$\displaystyle {\hat {f}}(\omega )=$

$\displaystyle {\hat {f}}(\nu )=$

$\displaystyle 1$

$\displaystyle \delta (\xi )$

$\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )$

$\displaystyle 2\pi \delta (\nu )$

$\displaystyle \delta (x)\,$

$\displaystyle 1$

$\displaystyle {\frac {1}{\sqrt {2\pi }}}\,$

$\displaystyle 1$

$\displaystyle e^{iax}$

$\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)$

$\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)$

$\displaystyle 2\pi \delta (\nu -a)$

$\displaystyle \cos(ax)$

$\displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}$

$\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\,$

$\displaystyle \pi \left(\delta (\nu -a)+\delta (\nu +a)\right)$

$\textstyle \cos(ax)=$

$\displaystyle \sin(ax)$

$\displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}$

$\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}$

$\displaystyle -i\pi \left(\delta (\nu -a)-\delta (\nu +a)\right)$

$\textstyle \sin(ax)=$

$\displaystyle \cos(ax^{2})$

$\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)$

$\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$

$\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\nu ^{2}}{a}}-{\frac {\pi }{4}}\right)$

$\displaystyle \sin(ax^{2})\,$

$\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)$

$\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$

$\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\nu ^{2}}{a}}-{\frac {\pi }{4}}\right)$

$\displaystyle x^{n}\,$

$\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\,$

$\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,$

$\displaystyle 2\pi i^{n}\delta ^{(n)}(\nu )\,$

$\textstyle \delta ^{(n)}(\xi )$

$\displaystyle {\frac {1}{x}}$

$\displaystyle -i\pi \operatorname {sgn}(\xi )$

$\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )$

$\displaystyle -i\pi \operatorname {sgn}(\nu )$

$\displaystyle {\frac {1}{x^{n}}}:=$

$\displaystyle -i\pi {\frac {(-2\pi i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )$

$\displaystyle -i{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )$

$\displaystyle -i\pi {\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )$

$\textstyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|$

$\displaystyle |x|^{\alpha }\,$

$\displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}$

$\displaystyle {\frac {-2}{\sqrt {2\pi }}}{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}$

$\displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}$

$|x|^{\alpha }$

$\textstyle \alpha \mapsto |x|^{\alpha }$

$|x|^{\alpha }$

${\frac {1}{\sqrt {|x|}}}\,$

${\frac {1}{\sqrt {|\xi |}}}$

${\frac {1}{\sqrt {|\omega |}}}$

${\frac {\sqrt {2\pi }}{\sqrt {|\nu |}}}$

$\displaystyle \operatorname {sgn}(x)$

$\displaystyle {\frac {1}{i\pi \xi }}$

$\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }}$

$\displaystyle {\frac {2}{i\nu }}$

$\displaystyle u(x)$

$\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)$

$\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)$

$\displaystyle \pi \left({\frac {1}{i\pi \nu }}+\delta (\nu )\right)$

$\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)$

$\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)$

$\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)$

$\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {2\pi k}{T}}\right)$

$\sum _{n=-\infty }^{\infty }e^{inx}=$

$\displaystyle J_{0}(x)$

$\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}$

$\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$

$\displaystyle {\frac {2\,\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$

$\displaystyle J_{n}(x)$

$\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}$

$\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$

$\displaystyle {\frac {2(-i)^{n}T_{n}(\nu )\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$

$\displaystyle \log \left|x\right|$

$\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)$

$\displaystyle -{\frac {\sqrt {\pi /2}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right)$

$\displaystyle -{\frac {\pi }{\left|\nu \right|}}-2\pi \gamma \delta \left(\nu \right)$

$\gamma$

$\displaystyle \left(\mp ix\right)^{-\alpha }$

$\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}$

$\displaystyle {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}$

$\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \nu \right)\left(\pm \nu \right)^{\alpha -1}$

$\displaystyle f(x,y)$

$\displaystyle {\hat {f}}(\xi _{x},\xi _{y})=$

$\displaystyle {\hat {f}}(\omega _{x},\omega _{y})=$

$\displaystyle {\hat {f}}(\nu _{x},\nu _{y})=$

$\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}$

$\displaystyle {\frac {1}{|ab|}}e^{-\pi \left(\xi _{x}^{2}/a^{2}+\xi _{y}^{2}/b^{2}\right)}$

$\displaystyle {\frac {1}{2\pi \cdot |ab|}}e^{\frac {-\left(\omega _{x}^{2}/a^{2}+\omega _{y}^{2}/b^{2}\right)}{4\pi }}$

$\displaystyle {\frac {1}{|ab|}}e^{\frac {-\left(\nu _{x}^{2}/a^{2}+\nu _{y}^{2}/b^{2}\right)}{4\pi }}$

$\displaystyle \mathrm {circ} ({\sqrt {x^{2}+y^{2}}})$

$\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}$

$\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}$

$\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}$

$\displaystyle f(\mathbf {x} )\,$

$\displaystyle {\hat {f}}({\boldsymbol {\xi }})=$

$\displaystyle {\hat {f}}({\boldsymbol {\omega }})=$

$\displaystyle {\hat {f}}({\boldsymbol {\nu }})=$

$\displaystyle \chi _{[0,1]}(|\mathbf {x} |)(1-|\mathbf {x} |^{2})^{\delta }$

$\displaystyle \pi ^{-\delta }\Gamma (\delta +1)|{\boldsymbol {\xi }}|^{-n/2-\delta }$

$\displaystyle \times J_{n/2+\delta }(2\pi |{\boldsymbol {\xi }}|)$

$\displaystyle 2^{-\delta }\Gamma (\delta +1)\left|{\boldsymbol {\omega }}\right|^{-n/2-\delta }$

$\displaystyle \times J_{n/2+\delta }(|{\boldsymbol {\omega }}|)$

$\displaystyle \pi ^{-\delta }\Gamma (\delta +1)\left|{\frac {\boldsymbol {\nu }}{2\pi }}\right|^{-n/2-\delta }$

$\displaystyle \times J_{n/2+\delta }(|{\boldsymbol {\nu }}|)$

$\displaystyle |\mathbf {x} |^{-\alpha },\quad 0<\operatorname {Re} \alpha <n.$

$\displaystyle c_{n-\alpha ,n}(2\pi )^{\alpha -n}|{\boldsymbol {\xi }}|^{-(n-\alpha )}$

$\displaystyle c_{n-\alpha ,n}(2\pi )^{-n/2}|{\boldsymbol {\omega }}|^{-(n-\alpha )}$

$\displaystyle c_{n-\alpha ,n}|{\boldsymbol {\nu }}|^{-(n-\alpha )}$

$\displaystyle {\frac {1}{\left\|{\boldsymbol {\sigma }}\right\|\left(2\pi \right)^{n/2}}}e^{-{\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }{\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}\mathbf {x} }$

$\displaystyle e^{-{\frac {1}{2}}{\boldsymbol {\nu }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\nu }}}$

$\displaystyle e^{-2\pi \alpha |\mathbf {x} |}$

$c_{n}{\frac {\alpha }{(\alpha ^{2}+|\xi |^{2})^{(n+1)/2}}}$

$c_{\alpha ,n}=\pi ^{n/2}2^{\alpha }{\frac {\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}}$

${\boldsymbol {\Sigma }}={\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }$

${\boldsymbol {\Sigma }}^{-1}={\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}$

$c_{n}=\Gamma ((n+1)/2)/\pi ^{(n+1)/2}$