# Multiple 1D gaussian filter

Given

$$\begin{cases} &f_0(x)=1 \\ &f_{n+1}(x)= (\varphi*(f\mathbb{I}_{[a_n,b_n]}))(x)=\int_{-\infty}^{+\infty}\varphi(x-t)f_n(t)\mathbb{I}_{[a_n,b_n]}(t)dt = \int_{a_n}^{b_n}\varphi(x-t)f_n(t)dt \quad n = 0,...,N-1 \\ \end{cases}$$ with $$\{a_n\}_{n=1,...,N},\{b_n\}_{n=1,...,N} \in \Bbb R$$ and $$\varphi(x)$$ the standard gaussian function $$\varphi(x) = \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2} \right)}$$

The functions $$f_{n+1}$$ are defined as a convolution of $$\varphi$$ and $$f_n\mathbb{I}_{[a_n,b_n]}$$ for $$n = 0,...,N-1$$.

The goal is to compute the value $$V$$ as quick as possible. $$V=\int_{a_N}^{b_N}f_N(x)dx$$

Numerical absolute error is on the order of $$\mathcal{O}(10^{-3})$$ is sufficient.

This is some kind of multiple 1D Gaussian blur or multiple Gaussian filters.

I wonder whether this problem is known in signal processing field?

• I'm seeking a numerical solution (a closed form solution, or even an approximate closed form solution, are better but I doubt that they exist). An acccuracy on the order of $\mathcal{O}(10^{-3})$ is sufficient. However, the algorithm should be as fast as possible because later I need to compute not less than one thousand million values $V$.