What is "boxcar function"

n. (context mathematics English) Any function whose value is zero except for a finite part of its argument for which it has a constant non-zero value

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. The boxcar function can be expressed in terms of the uniform distribution as

boxcar(x) = (b − a)A f(a, b; x) = A(H(x − a) − H(x − b)), 

where f(a,b;x) is the uniform distribution of x for the interval [a, b] and H(x) is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points. These values are probably best chosen for each individual application.

When a boxcar function is selected as the impulse response of a filter, the result is a moving average filter.

The function is named after its resemblance to a boxcar, a type of railroad car.