View source for Window function

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301401.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301402.png" /> is a suitable window function. Quite often, scaled and translated versions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301403.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301404.png" /> that equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301405.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301406.png" /> and zero elsewhere (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301408.png" /> it is arbitrarily defined, usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w1301409.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014010.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014011.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014012.png" />) is the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014013.png" /></td> </tr></table>

a version of the sinc function (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014016.png" />), see [[#References|[a2]]], pp. 61, 104. In terms of the Heaviside function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014017.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014022.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014023.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014024.png" /></td> </tr></table>

There is also a relation with the Dirac delta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014025.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130140/w13014026.png" /></td> </tr></table>

====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>