It is known that the convolution of two Gaussian function is also a scaled Gaussian function. This convolution is taken from $–\infty$ to $\infty$ since the Gaussian function has infinite support.
Convolutions are typically used as linear filters to smooth a function observed over some data points. An example is a data set where $x=[0,1,2,3,4,5], y=[10,13,16,14,30]$.
Typically people smooth out these data through a convolution with a kernel (Gaussian Kernel) using an infinite integral. Why don’t they only integrate over the space the data points are observed in. Ex integrate over $[0,5]$.