Convolution

Integrating a resolution function to find the observed function.

When finding a function $f(x)$ with a measuring instrument, e.g. position with a microscope, then the instrument has a set resolution to make the measurements.

A resolution function $g(z-x)$ is defined as the probability density that the true value in $(x,x+dx)$ is shifted to $(z,z+dz)$, i.e. blurring a measurement made.

Integrating a resolution function over the values of $x$ that could lead to a reading in $(z,z+dz)$ (i.e. that could be blurred), leads to the observed function, $h(z)={\int }_{-\infty }^{\infty }f(x)g(z-x)dx$ - this is called the convolution of $f$ and $g$.

Perfect resolution is given by the $\delta$ function $h(z)={\int }_{-\infty }^{\infty }f(x)\delta (z-x)dx=f(z)$; the observed function is equal to the true function.

Real resolution functions might be symmetric, i.e. blurred, or asymmetric, i.e. systematic error.

The convolution of $f$ and $g$ is sometimes written as $f\ast g$ and is such that $f\ast g=g\ast f$ (commutative property).

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Example One

Find the convolution of $e^{-x^2}$ with the sum of the $\delta$ functions $\delta (x-a)+\delta (x-b)$::$\exp (-(x-a{)}^2)+\exp (-(z-b{)}^2)$.