User:Physikerwelt/sandbox/FT


 * $$f(t)$$


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


 * $$\omega$$


 * $$g(t)$$


 * $$\scriptstyle f(t)$$


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


 * $$\scriptstyle g(t)$$


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


 * $$\scriptstyle t$$


 * $$\scriptstyle \omega$$


 * $$\scriptstyle t$$


 * $$\scriptstyle \omega$$


 * $$\hat{f}$$


 * $$f : \mathbb R \rightarrow \mathbb C$$


 * $$\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx,$$


 * $$f$$


 * $$\hat f$$


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


 * $$f$$


 * $$\hat f$$


 * $$f$$


 * $$\hat{f}$$


 * $$L^2$$


 * $$L^1$$


 * $$L^\infty$$


 * $$\hat{f}$$


 * $$c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x)\ e^{-2\pi i(n/T) x} \, dx.$$


 * $$c_n = (1/T)\hat f(n/T)$$


 * $$f(x)=\sum_{n=-\infty}^\infty c_n\ e^{2\pi i(n/T) x} =\sum_{n=-\infty}^\infty \hat{f}(\xi_n)\ e^{2\pi i\xi_n x}\Delta\xi,$$


 * $$\hat f(3)$$


 * $$\hat f(5)$$


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


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


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


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


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


 * $$h(x)=f(x-x_0),$$


 * $$\hat{h}(\xi)= e^{-i\,2\pi \,x_0\,\xi }\hat{f}(\xi).$$


 * $$h(x)=e^{i \, 2\pi \, x \,\xi_0}f(x),$$


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


 * $$h(x)=f(ax)$$


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


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


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


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


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


 * $$\hat{f}$$


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


 * $$\xi=0 $$


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


 * $$\xi=0$$


 * $$\hat{f}.$$


 * $$\mathcal{F},$$


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


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


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


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


 * $$\mathcal{P}$$


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


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


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


 * $$t$$


 * $$\xi$$


 * $$L^2(b R)$$(was L^2(\b R))


 * $$\xi$$


 * $$t$$


 * $$t$$


 * $$\xi$$


 * $$t$$


 * $$t$$


 * $$2\pi$$


 * $$\xi$$


 * $$t$$


 * $$\xi$$


 * $$t$$


 * $$t$$


 * $$\xi$$


 * $$\xi$$


 * $$t$$


 * $$t$$


 * $$\xi$$


 * $$\omega = 2\pi \xi$$


 * $$\hat x_1$$


 * $$\hat x$$


 * $$\hat x_1(\omega) = \hat x \left({\omega\over2\pi}\right) = \int_{-\infty}^\infty x(t) e^{-i\omega t} \, dt$$


 * $$ x(t) = {1\over{2\pi}}\int_{-\infty}^\infty \hat x_1(\omega) e^{it\omega} \, d\omega .$$


 * $$\sqrt{2\pi}$$


 * $$\hat x_2 (\omega) = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^\infty x(t) e^{-i\omega t} \, dt ,$$


 * $$ x(t) = {1\over\sqrt{2\pi}}\int_{-\infty}^\infty \hat x_2(\omega) e^{it\omega} \, d\omega .$$


 * $$i$$


 * $$i$$


 * $$i$$


 * $$-i$$


 * $$\phi$$


 * $$f$$


 * $$X$$


 * $$x$$


 * $$2\pi$$


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


 * $$\hat f$$


 * $$\|\hat{f}\|_{\infty}\leq \|f\|_1$$


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


 * $$\hat f$$


 * $$\hat f$$


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


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


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


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


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


 * $$ f$$


 * $$ \sum_n \hat f(n) = \sum_n f (n).$$


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


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


 * $$f(x)$$


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


 * $$|\xi|\to\infin$$


 * $$f(x)$$


 * $$|x|\to\infin$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


 * $$ \hat\psi_n(\xi) = (-i)^n {\psi}_n(\xi) $$


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


 * $$\xi$$


 * $$f$$


 * $$\xi = \sigma + i\tau$$


 * $$f$$


 * $$n$$


 * $$n$$


 * $$\hat f(\sigma + i\tau) $$


 * $$a>0$$


 * $$n\geq 0$$


 * $$ \vert \xi ^n \hat f(\xi) \vert \leq C e^{a\vert\tau\vert} $$


 * $$C$$


 * $$f$$


 * $$[-a,a]$$


 * $$\hat f$$


 * $$\sigma$$


 * $$\tau$$


 * $$\tau$$


 * $$\sigma$$


 * $$f$$


 * $$L^2$$


 * $$n=0$$


 * $$f$$


 * $$t \geq 0$$


 * $$f$$


 * $$\hat f$$


 * $$\tau < 0$$


 * $$ \tau $$


 * $$ \hat f (\xi) $$


 * $$ F (s) $$


 * $$s$$


 * $$f$$


 * $$f(t)$$


 * $$ \vert f(t) \vert < C e^{a\vert t\vert} $$


 * $$C,a \geq 0$$


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


 * $$2\pi \tau < -a$$


 * $$f$$


 * $$ F(s) = \int_0^\infty f(t) e^{-st} \, dt.$$


 * $$f$$


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


 * $$s=2\pi i \xi$$


 * $$\hat f$$


 * $$a\leq \tau \leq b$$


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


 * $$f(t) = 0$$


 * $$t < 0$$


 * $$\vert f(t) \vert < C e^{a\vert t\vert} $$


 * $$C,a > 0$$


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


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


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


 * $$b > a$$


 * $$F(s)$$


 * $$f(t)$$


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


 * $$ L^1$$


 * $$ t$$


 * $$ f$$


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


 * $$\left\langle \mathbf x, \boldsymbol \xi \right\rangle$$


 * $$\hat f(\xi)$$


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


 * $$\hat f(\xi)$$


 * $$D_0(f)=\int_{-\infty}^\infty x^2|f(x)|^2\,dx.$$


 * $$D_0(f)D_0(\hat{f}) \geq \frac{1}{16\pi^2}$$


 * $$f(x)=C_1 \, e^{-\pi x^2/\sigma^2}$$


 * $$\hat{f}(\xi)= \sigma C_1 \, e^{-\pi\sigma^2\xi^2}$$


 * $$C_1 = \sqrt[4]{2} / \sqrt{\sigma}$$


 * $$\left(\int_{-\infty}^\infty (x-x_0)^2|f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty(\xi-\xi_0)^2|\hat{f}(\xi)|^2\,d\xi\right)\geq \frac{1}{16\pi^2}$$


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


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


 * $$f$$


 * $$\lambda$$


 * $$a$$


 * $$b$$


 * $$a$$


 * $$b$$


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


 * $$\hat{f}(\xi)=F_0(|\xi|)P(\xi)$$


 * $$F_0(r)=2\pi i^{-k}r^{-(n+2k-2)/2}\int_0^\infty f_0(s)J_{(n+2k-2)/2}(2\pi rs)s^{(n+2k)/2}\,ds.$$


 * $$f_R(x) = \int_{E_R}\hat{f}(\xi) e^{2\pi ix\cdot\xi}\, d\xi, \quad x \in \mathbf{R}^n.$$


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


 * $$\mathcal{F}$$


 * $$\hat{f}(\xi) = \lim_{R\to\infty}\int_{|x|\le R} f(x) e^{-2\pi i x\cdot\xi}\,dx$$


 * $$\mathcal{F}$$


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


 * $$T_f(\varphi)=\int_{\mathbf{R}^n}f(x)\varphi(x)\,dx$$


 * $$\hat{T}_f$$


 * $$\hat{T}_f (\varphi)= T_f(\hat{\varphi})$$


 * $$\hat\mu(\xi)=\int_{\mathbf{R}^n} \mathrm{e}^{-2\pi i x \cdot \xi}\,d\mu.$$


 * $$\hat G$$


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


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


 * $$\hat G$$


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


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


 * $$\langle \hat{\mu}\xi,\eta\rangle_{H_\sigma} = \int_G \langle \overline{U}^{(\sigma)}_g\xi,\eta\rangle\,d\mu(g)$$


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


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


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


 * $$\|E\| = \sup_{\sigma\in\Sigma}\|E_\sigma\|$$


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


 * $$f(g) = \sum_{\sigma\in\Sigma} d_\sigma \operatorname{tr}(\hat{f}(\sigma)U^{(\sigma)}_g)$$


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


 * $$f$$


 * $$g$$


 * $$y$$


 * $$\hat y$$


 * $$\hat y$$


 * $$y$$


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


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


 * $$a_+$$


 * $$a_-$$


 * $$b_+$$


 * $$b_-$$


 * $$a_\pm$$


 * $$b_\pm$$


 * $$x$$


 * $$a_\pm$$


 * $$b_\pm$$


 * $$y$$


 * $$t=0$$


 * $$t=0$$


 * $$x$$


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


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


 * $$y$$


 * $$t$$


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


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


 * $$a_\pm$$


 * $$b_\pm$$


 * $$\xi$$


 * $$\xi$$


 * $$f$$


 * $$g$$


 * $$a_\pm$$


 * $$b_\pm$$


 * $$f$$


 * $$g$$


 * $$x$$


 * $$t$$


 * $$\hat y$$


 * $$y(x,t)$$


 * $$L^1$$


 * $$x$$


 * $$2\pi i\xi$$


 * $$t$$


 * $$2\pi if$$


 * $$f $$


 * $$\hat y$$


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


 * $$\hat y(\xi,f) = 0$$


 * $$\xi = \pm f$$


 * $$\hat f = \delta(\xi\pm f)$$


 * $$\xi^2-f^2 = 0$$


 * $$\xi = f$$


 * $$\xi = -f$$


 * $$\phi$$


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


 * $$s_+$$


 * $$ s_-$$


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


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


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


 * $$x$$


 * $$x$$


 * $$s_\pm$$


 * $$L^1$$


 * $$L^2$$


 * $$q$$


 * $$p$$


 * $$q$$


 * $$p$$


 * $$p$$


 * $$q$$


 * $$p$$


 * $$q$$


 * $$p$$


 * $$q$$


 * $$q$$


 * $$p$$


 * $$L^2$$


 * $$L^2$$


 * $$q$$


 * $$p$$


 * $$i$$


 * $$V(x)$$


 * $$\psi$$


 * $$t=0$$


 * $$R$$


 * $$f$$


 * $$\tau$$


 * $$f$$


 * $$f$$


 * $$R$$


 * $$ \tau$$


 * $$ \tau = $$


 * $$f$$


 * $$f$$


 * $$f(t)$$


 * $$t$$


 * $$f$$


 * $$f$$


 * $$P$$


 * $$ \xi$$


 * $$\hat f(\xi)$$


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


 * $$\hat f(\xi)$$


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


 * $$A(\xi) = |\hat f(\xi)|,$$


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


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


 * $$\mathcal F$$


 * $$\mathcal F(f)$$


 * $$\mathcal F$$


 * $$\mathcal F f$$


 * $$\mathcal F(f)$$


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


 * $$(\mathcal F f)(\xi)$$


 * $$\mathcal F$$


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


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


 * $$\mathcal F(f(x + x_0)) = \mathcal F(f(x)) e^{2\pi i \xi x_0}$$


 * $$\omega = 2\pi \xi,$$


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


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


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


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


 * $$\displaystyle \hat{f}_1(\xi)\ \stackrel{\mathrm{def}}{=}\ \int_{\mathbf{R}^n} f(x) e^{-2 \pi i x\cdot\xi}\, dx = \hat{f}_2(2 \pi \xi)=(2 \pi)^{n/2}\hat{f}_3(2 \pi \xi)$$


 * $$\displaystyle \hat{f}_3(\omega) \stackrel{\mathrm{def}}{{}={}} \frac{1}{(2 \pi)^{n/2}} \int_{\mathbf{R}^n} f(x) e^{-i \omega\cdot x}\, dx = \frac{1}{(2 \pi)^{n/2}} \hat{f}_1 \! \left(\frac{\omega}{2 \pi}\right) = \frac{1}{(2 \pi)^{n/2}} \hat{f}_2(\omega)$$


 * $$\displaystyle \hat{f}_2(\omega) \ \stackrel{\mathrm{def}}{=}\int_{\mathbf{R}^n} f(x) e^{-i\omega\cdot x}\, dx = \hat{f}_1 \! \left(\frac{\omega}{2 \pi}\right) = (2 \pi)^{n/2} \hat{f}_3(\omega)$$


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


 * $$\hat{f}$$


 * $$\hat{g}$$


 * $$\hat{h}$$


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


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


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


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


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


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


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


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


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


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


 * $$\displaystyle e^{- i a \omega} \hat{f}(\omega)\,$$


 * $$\displaystyle e^{- i a \nu} \hat{f}(\nu)\,$$


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


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


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


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


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


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


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


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


 * $$\displaystyle |a|\,$$


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


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


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


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


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


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


 * $$\hat{f}$$


 * $$x$$


 * $$\xi$$


 * $$\omega$$


 * $$\nu$$


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


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


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


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


 * $$\displaystyle x^n f(x)\,$$


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


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


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


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


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


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


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


 * $$\displaystyle f * g\,$$


 * $$f$$


 * $$g$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


 * $$\displaystyle f(x)$$


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


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


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


 * $$\displaystyle \operatorname{rect}(a x) \,$$


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


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


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


 * $$\displaystyle \operatorname{sinc}(a x)\,$$


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


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


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


 * $$\displaystyle \operatorname{sinc}^2 (a x)$$


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


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


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


 * $$\displaystyle \operatorname{tri} (a x)$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


 * $$\displaystyle \operatorname{sech}(a x) \,$$


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


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


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


 * $$\displaystyle e^{-\frac{a^2 x^2}2} H_n(a x)\,$$


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


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


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


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


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


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


 * $$H_n$$


 * $$\displaystyle f(x)$$


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


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


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


 * $$\displaystyle 1$$


 * $$\displaystyle \delta(\xi)$$


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


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


 * $$\displaystyle \delta(x)\,$$


 * $$\displaystyle 1$$


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


 * $$\displaystyle 1$$


 * $$\displaystyle e^{i a x}$$


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


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


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


 * $$\displaystyle \cos (a x)$$


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


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


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


 * $$\textstyle \cos(a x) = $$


 * $$\displaystyle \sin( ax)$$


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


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


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


 * $$\textstyle \sin(a x) = $$


 * $$\displaystyle \cos ( a x^2 ) $$


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


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


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


 * $$\displaystyle \sin ( a x^2 ) \,$$


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


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


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


 * $$\displaystyle x^n\,$$


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


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


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


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


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


 * $$\displaystyle -i\pi\sgn(\xi)$$


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


 * $$\displaystyle -i\pi\sgn(\nu)$$


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


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


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


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


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


 * $$\displaystyle |x|^\alpha\,$$


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


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


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


 * $$|x|^\alpha$$


 * $$\textstyle \alpha\mapsto |x|^\alpha$$


 * $$|x|^\alpha$$


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


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


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


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


 * $$\displaystyle \sgn(x)$$


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


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


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


 * $$\displaystyle u(x)$$


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


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


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


 * $$\displaystyle \sum_{n=-\infty}^{\infty} \delta (x - n T)$$


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


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


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


 * $$\sum_{n=-\infty}^{\infty} e^{inx}=$$


 * $$\displaystyle J_0 (x)$$


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


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


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


 * $$\displaystyle J_n (x)$$


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


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


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


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


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


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


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


 * $$\gamma$$


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


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


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


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


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


 * $$\displaystyle \hat{f}(\xi_x, \xi_y)=$$


 * $$\displaystyle \hat{f}(\omega_x,\omega_y)=$$


 * $$\displaystyle \hat{f}(\nu_x,\nu_y)=$$


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


 * $$\displaystyle \frac{1}{|ab|} e^{-\pi\left(\xi_x^2/a^2 + \xi_y^2/b^2\right)}$$


 * $$\displaystyle \frac{1}{2\pi\cdot|ab|} e^{\frac{-\left(\omega_x^2/a^2 + \omega_y^2/b^2\right)}{4\pi}}$$


 * $$\displaystyle \frac{1}{|ab|} e^{\frac{-\left(\nu_x^2/a^2 + \nu_y^2/b^2\right)}{4\pi}}$$


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


 * $$\displaystyle \frac{J_1\left(2 \pi \sqrt{\xi_x^2+\xi_y^2}\right)}{\sqrt{\xi_x^2+\xi_y^2}}$$


 * $$\displaystyle \frac{J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}$$


 * $$\displaystyle \frac{2\pi J_1\left(\sqrt{\nu_x^2+\nu_y^2}\right)}{\sqrt{\nu_x^2+\nu_y^2}}$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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