It is a well known fact in mathematics that the product of two Gaussian functions is also a Gaussian function. In this post I want to present this result for future reference for me and for anyone who might find it useful.
The product theorem for Gaussian functions states that the product of two overlapping Gaussian functions is also a Gaussian function and determines the center and width of the resulting function in terms of the parameters of the two original functions.
To illustrate the identity, consider a Gaussian function of the form , where , and , correspond to its center, width and amplitude, respectively. The result of the product of two Gaussian functions and with different amplitudes, widths and central positions, , is equal to
where the centroid is given by
and the width by
The theorem shows that the result of the product of two Gaussian functions is a new Gaussian function of width , centered in the position , whose amplitude strongly depends on the factor . In addition, the resulting function is narrower than either of the two original Gaussian functions, and its center lies within the interval . Notice that the above result can be further simplified if we consider the scenario where the Gaussian functions are centered in different positions but have the same width. If we define , the centroid simplies to , whereas the width is given by .
The next figures show two representative cases where two different Gaussian functions with different and equal widths are multiplied. In both cases the result is a Gaussian function narrower than the original functions centered in between the two original centroids, and .
- E. Wolf, J. T. Foley and F. Gori, "Frequency shifts of spectral lines produced by scattering from spatially random media," J. Opt. Soc. Am. A, 7, 173, (1990)