Box smoothing of a two-sided exponential ($w(x)=(2a{)}^{-1}{\int }_{-a}^a(2b{)}^{-1}{e}^{-|x-t|/b}dt$)

For $a,b>0$, the convolution of the normalized interval indicator $(2a{)}^{-1}{1}_{[-a,a]}$ with the two-sided exponential $(2b{)}^{-1}{e}^{-|x|/b}$ isSplit the defining integral at $t=x$ in the interior case; outside the interval the absolute value has a fixed sign. These elementary integrals prove the formula and its continuity at the joining points. With Fourier transform convention $\hat{w}(k)=\int w(x)e^{-ikx}dx$, the convolution theorem gives $\hat{w}(k)=\sin (ak)/[ak(1+b^2{k}^2)]$, using its continuous value at $k=0$. Parseval's identity can therefore turn the squared integral of this rational-trigonometric transform into a piecewise elementary real-space integral.