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) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

The horizontal Centroids add

(10) |

(11) |

(12) |

**References**

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.

© 1996-9

1999-05-25