convolution

A convolution is a type of product function of two functions defined as:

(f*g)(t) means ∫ f(x)g(t-x) dx

   (for x from -∞ to ∞)

f*g - the convolution of functions f and g.

One way to explain it is that it is the amount of overlap when one function is shifted over another.

Convolutions are the means of producing density functions that result from combination of arbitrary probability density functions. It can be appropriate for combining individual density functions that model aspects contributing to some physical process, such as those modeling spectral line shapes. The Voigt function is a convolution of two functions (for two processes) affecting spectral-line shape (the Voigt profile). Convolution appears to be basic to some optical applications and gives a mathematical model for effects of blurring, e.g., from aberration, and also the effect of detectors on images. It models the effect of a function representing the intensity of incoming light with another function representing the distortion, and the mathematics of convolutions spells out the possibility and means of reversing the distortion. It also yields the result of some particular spectral energy distribution (SED) sensed with some particular sensitivity function.

In observed processes or observation techniques involving implicit convolution, a step in analysis may carrying out the inverse mathematical process, deconvolution. Other than "take a shot at it" guesses, there are mathematical methods to assist the process, which is one use of the Fourier transform. In addition to spectrography, deconvolution is used in parsing of observation data in interferometry and aperture synthesis.