Editing Fourier transform (section)

=== Uncertainty principle ===
{{Further|Uncertainty principle}}

Generally speaking, the more concentrated {{math|''f''(''x'')}} is, the more spread out its Fourier transform {{math|''f̂''(''ξ'')}} must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in {{mvar|x}}, its Fourier transform stretches out in {{mvar|ξ}}. It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an [[uncertainty principle]] by viewing a function and its Fourier transform as [[conjugate variables]] with respect to the [[symplectic form]] on the [[time–frequency representation|time–frequency domain]]: from the point of view of the [[linear canonical transformation]], the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the [[Symplectic vector space|symplectic form]].

Suppose {{math|''f''(''x'')}} is an integrable and [[square-integrable]] function. Without loss of generality, assume that {{math|''f''(''x'')}} is normalized:
<math display="block">\int_{-\infty}^\infty |f(x)|^2 \,dx=1.</math>

It follows from the [[Plancherel theorem]] that {{math|''f̂''(''ξ'')}} is also normalized.

The spread around {{math|''x'' {{=}} 0}} may be measured by the ''dispersion about zero'' defined by<ref>{{harvnb|Pinsky|2002|loc=chpt. 2.4.3 The Uncertainty Principle}}</ref>
<math display="block">D_0(f)=\int_{-\infty}^\infty x^2|f(x)|^2\,dx.</math>

In probability terms, this is the [[Moment (mathematics)|second moment]] of {{math|{{abs|''f''(''x'')}}<sup>2</sup>}} about zero.

The uncertainty principle states that, if {{math|''f''(''x'')}} is absolutely continuous and the functions {{math|''x''·''f''(''x'')}} and {{math|''f''{{′}}(''x'')}} are square integrable, then
<math display="block">D_0(f)D_0(\hat{f}) \geq \frac{1}{16\pi^2}.</math>

The equality is attained only in the case
<math display="block">\begin{align} f(x) &= C_1 \, e^{-\pi \frac{x^2}{\sigma^2} }\\
\therefore \hat{f}(\xi) &= \sigma C_1 \, e^{-\pi\sigma^2\xi^2} \end{align} </math>
where {{math|''σ'' > 0}} is arbitrary and {{math|1=''C''<sub>1</sub> = {{sfrac|{{radic|2|4}}|{{sqrt|''σ''}}}}}} so that {{mvar|f}} is {{math|''L''<sup>2</sup>}}-normalized. In other words, where {{mvar|f}} is a (normalized) [[Gaussian function]] with variance {{math|''σ''<sup>2</sup>/2{{pi}}}}, centered at zero, and its Fourier transform is a Gaussian function with variance {{math|''σ''<sup>−2</sup>/2{{pi}}}}.  Gaussian functions are examples of [[Schwartz function]]s (see the discussion on tempered distributions below). 

In fact, this inequality implies that:
<math display="block">\left(\int_{-\infty}^\infty (x-x_0)^2|f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty(\xi-\xi_0)^2\left|\hat{f}(\xi)\right|^2\,d\xi\right)\geq \frac{1}{16\pi^2}, \quad \forall x_0, \xi_0 \in \mathbb{R}.</math>
In [[quantum mechanics]], the [[momentum]] and position [[wave function]]s are Fourier transform pairs, up to a factor of the [[Planck constant]]. With this constant properly taken into account, the inequality above becomes the statement of the [[Heisenberg uncertainty principle]].<ref>{{harvnb|Stein|Shakarchi|2003|loc= chpt. 5.4 The Heisenberg uncertainty principle}}</ref>

A stronger uncertainty principle is the [[Hirschman uncertainty|Hirschman uncertainty principle]], which is expressed as:
<math display="block">H\left(\left|f\right|^2\right)+H\left(\left|\hat{f}\right|^2\right)\ge \log\left(\frac{e}{2}\right)</math>
where {{math|''H''(''p'')}} is the [[differential entropy]] of the [[probability density function]] {{math|''p''(''x'')}}:
<math display="block">H(p) = -\int_{-\infty}^\infty p(x)\log\bigl(p(x)\bigr) \, dx</math>
where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.