تبدیل فوریه
جدول تبدیل‌های فوریه مهم > توابع انتگرال‌گیری شونده مربعی، یک بعدی

تبدیل‌های فوریه زیر را می‌توان در Campbell & Foster (1948)، Erdélyi (1954)، یا Kammler (2000، قسمت ضمیمه) یافت.
|  | تابع | تبدیل فوریه فرکانس واحد، عادی | تبدیل فوریه فرکانس واحد، زاویه ای | تبدیل فوریه فرکانس زاویه ای غیر واحد | ملاحظات |
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|  |  f(x)\, |  {\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx\end{aligned}} |  {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}} |  {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}} |  |
|  ۲۰۱ |  \operatorname {rect} (ax)\, |  {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right) |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right) | The rectangular pulse and the normalized sinc function, here defined as sinc(x) = sin(πx)/πx |
| ۲۰۲ |  \operatorname {sinc} (ax)\, |  {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right) | Dual of rule 201. The rectangular function is an ideal low-pass filter, and the sinc function is the non-causal impulse response of such a filter. The sinc function is defined here as sinc(x) = sin(πx)/πx |
| ۲۰۳ |  \operatorname {sinc} ^{2}(ax) |  {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right) |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right) | The function tri(x) is the triangular function |
| ۲۰۴ |  \operatorname {tri} (ax) |  {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\, |  {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right) |  {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right) | Dual of rule 203. |
| ۲۰۵ |  e^{-ax}u(x)\, |  {\frac {1}{a+2\pi i\xi }} |  {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}} |  {\frac {1}{a+i\nu }} | The function u(x) is the Heaviside unit step function and a> 0. |
| ۲۰۶ |  e^{-\alpha x^{2}}\, |  {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi \xi )^{2}}{\alpha }}} |  {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}} |  {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha }}} | This shows that, for the unitary Fourier transforms, the Gaussian function e−αx2 is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α)> ۰. |
| ۲۰۷ |  e^{-i\alpha x^{2}}\, |  {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{i({\frac {(\pi \xi )^{2}}{\alpha }}-{\frac {\pi }{4}})} |  {\frac {1}{\sqrt {2\alpha }}}\cdot e^{i({\frac {\omega ^{2}}{4\alpha }}-{\frac {\pi }{4}})} |  {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{i({\frac {\nu ^{2}}{4\alpha }}-{\frac {\pi }{4}})} | This is known as the complex quadratic-phase sinusoid, or the "chirp" function. |
| ۲۰۸ |  \operatorname {e} ^{-a|x|}\, |  {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}} |  {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}} |  {\frac {2a}{a^{2}+\nu ^{2}}} | For Re(a)> 0. That is, the Fourier transform of a two-sided decaying exponential function is a Lorentzian function. |
| ۲۰۹ |  \operatorname {sech} (ax)\, |  {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right) |  {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right) |  {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right) | Hyperbolic secant is its own Fourier transform |
| ۲۱۰ |  e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\, |  {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right) |  {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right) |  {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{a}}\right) | Hn is the nth-order Hermite polynomial. If a = 1 then the Gauss–Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for n = ۰. |