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A function used to restrict consideration of an arbitrary function or signal in some way. The terms time-frequency localization, time localization or frequency localization are often used in this context. For instance, the windowed Fourier transform is given by

\begin{equation*} ( F _ { \text{win} } f ) ( \omega , t ) = \int f ( s ) g ( s - t ) e ^ { - i \omega s } d s, \end{equation*}

where $g ( t )$ is a suitable window function. Quite often, scaled and translated versions of $g ( t )$ are considered at the same time, [[#References|[a1]]], [[#References|[a3]]]. An example is the [[Gabor transform|Gabor transform]]. (See also [[Balian–Low theorem|Balian–Low theorem]]; [[Calderón-type reproducing formula|Calderón-type reproducing formula]].) Such window functions are also used in numerical analysis.

More specifically, the phrase window function refers to the function $r ( t )$ that equals $1$ on the interval $( - 1,1 )$ and zero elsewhere (at $- 1$ and $+ 1$ it is arbitrarily defined, usually $1/2$ or $0$). This function, as well as its scaled and translated versions, is also called the rectangle function or pulse function [[#References|[a2]]], pp. 30, 35, 60, 61. However, the phrase  "pulse function"  is also sometimes used for the [[Delta-function|delta-function]], see also [[Transfer function|Transfer function]].

The [[Fourier transform|Fourier transform]] of the specific rectangle function $r ( t )$ (with $r ( \pm 1 ) = 1 / 2$) is the function

\begin{equation*} g ( y ) = \left\{ \begin{array} { l l } { \frac { 1 } { \pi y } \operatorname { sin } 2 \pi y , } &amp; { y \neq 0, } \\ { 2 , } &amp; { y = 0, } \end{array} \right. \end{equation*}

a version of the sinc function ($\operatorname{sinc}( 0 ) = 1$, $\operatorname { sinc } ( x ) = x ^ { - 1 } \operatorname { sin } x$ for $x \neq 0$), see [[#References|[a2]]], pp. 61, 104. In terms of the Heaviside function $H ( x )$ ($H ( x ) = 0$ for $x &lt; 0$, $H ( 0 ) = 1 / 2$, $H ( x ) = 1$ for $x &gt; 0$), $r ( x )$ is given by

\begin{equation*} r ( x ) = H ( x + 1 ) - H ( x - 1 ). \end{equation*}

There is also a relation with the Dirac delta-function $\delta ( x )$:

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \frac { n } { 2 } r ( n x ) = \delta ( x ). \end{equation*}

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Daubechies,   "Ten lectures on wavelets" , SIAM  (1992)  pp. Chap. 1</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D.C. Champeney,   "A handbook of Fourier transforms" , Cambridge Univ. Press  (1989)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A.I. Saichev,   W.A. Woyczyński,   "Distributions in the physical and engineering sciences" , '''1: Distribution and fractal calculus, integral transforms and wavelets''' , Birkhäuser  (1997)  pp. 195ff</td></tr></table>