{{Short description|Mathematical function resembling a boxcar}} right|thumb|250px|A graphical representation of a boxcar function In mathematics, a '''boxcar function''' is any function which is zero over the entire real line except for a single interval where it is equal to a constant, ''A''.<ref>{{cite web| last=Weisstein|first=Eric W.|title=Boxcar Function|url=http://mathworld.wolfram.com/BoxcarFunction.html| publisher=MathWorld| accessdate=13 September 2013}}</ref> The function is named after its graph's resemblance to a boxcar, a type of railroad car. The boxcar function can be expressed in terms of the uniform distribution as <math display="block">\operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)),</math> where {{math|''f''(''a'',''b'';''x'')}} is the uniform distribution of ''x'' for the interval {{closed-closed|''a'', ''b''}} and <math>H(x)</math> is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points, which are usually best chosen depending on the individual application.
When a boxcar function is selected as the impulse response of a filter, the result is a simple moving average filter, whose frequency response is a sinc-in-frequency, a type of low-pass filter.
==See also== * Boxcar averager * Rectangular function * Step function * Top-hat filter ==References== {{reflist}}
{{mathanalysis-stub}} Category:Special functions