## Convolution

A convolution is an integral which expresses the amount of overlap of one function as it is shifted over another function . It therefore blends'' one function with another. For example, in synthesis imaging, the measured Dirty Map is a convolution of the true'' CLEAN Map with the Dirty Beam (the Fourier Transform of the sampling distribution). The convolution is sometimes also known by its German name, Faltung (folding''). A convolution over a finite range is given by (1)

where the symbol (occasionally also written as ) denotes convolution of and . Convolution is more often taken over an infinite range, (2)

Let , , and be arbitrary functions and a constant. Convolution has the following properties: (3) (4) (5) (6)

The Integral identity (7)

also gives a convolution. Taking the Derivative of a convolution gives (8)

The Area under a convolution is the product of areas under the factors,       (9) (10)

as do the Variances (11)

where (12)

References Convolution

Bracewell, R. Convolution.'' Ch. 3 in The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 25-50, 1965.

Hirschman, I. I. and Widder, D. V. The Convolution Transform. Princeton, NJ: Princeton University Press, 1955.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 464-465, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Convolution and Deconvolution Using the FFT.'' §13.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 531-537, 1992.