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Fast computation of convolution integral of a gaussian function

Given a convolution integral $$ g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right … Perhaps there exists a clever method using the property that $\varphi$ is a Gaussian function and/or $g(y)$ is a convolution of two function $\phi$ and $f\cdot \mathbb{I}_{[a,b]}$. …
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