# Analytical Approximation of A Convolution

Consider the function $f(x)=\frac{1}{\sqrt{2 \pi} s |x|} e^{-\frac{(a x +b )^2}{2 (s x)^2}}$, where $s>0$. I am interested in the convolution of $f$ with the Gaussian kernel $g(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{x^2}{2}}$. Let's refer to this convolution as $h(x)$,

$h(x)=\int_\mathbb{R} g(x-t) \, f(t) \, dt$.

I first tried to obtain a closed form expression for $h(x)$, but I failed. So my next goal is find an approximation to $h(x)$. However, that turns out to be challenging too. I appreciate any idea how to obtain an "analytical" approximation to $h(x)$.

Best

Golabi

• Thank you John. Integrating $\frac{1}{|t|}$ does not converge around zero as you said, but I do not understand why you eliminated the exp in front of $\frac{1}{|t|}$. More precisely it seems the full $f$ is well-behaved because $\lim_{t \rightarrow 0} f(t) = 0$. – Golabi Feb 9 '17 at 19:56
• Ok. I made a mistake previously. – John Feb 10 '17 at 1:43