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A convolution is an integral that 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").

Convolution is implemented in the Wolfram Language as Convolve[f, g, x, y] and DiscreteConvolve[f, g, n, m].

Convolution defines a product on the algebra of Schwartz functions on . For functions and that vanish for negative arguments, their convolution for reduces to the finite interval and is given by

where denotes the value of the convolution of and at , provided the integral exists.

More generally, convolution is taken over the entire real line,

(Bracewell 1965, p. 25), when the integral exists. The convolution of and is also occasionally written as .

The animations above graphically illustrate the convolution of two boxcar functions (left) and two Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a function of , the position indicated by the vertical green line. The gray region indicates the product as a function of for fixed , so its area as a function of is precisely the convolution. One feature to emphasize and which is not conveyed by these illustrations (since they both exclusively involve symmetric functions) is that the function must be mirrored before lagging it across and integrating.

The convolution of two boxcar functions and has the particularly simple form, for and ,

where is the Heaviside step function. The convolution of two Gaussians, with and ,

is another Gaussian

For the following algebraic and differentiation identities, let , , and be Schwartz functions on and a constant. Convolution satisfies the properties

(Bracewell 1965, p. 27), as well as

(Bracewell 1965, p. 49).

Taking the derivative of a convolution gives

(Bracewell 1965, p. 119).

For Lebesgue-integrable functions and , the integral of their convolution is the product of their integrals,

For nonnegative functions and with finite second raw moments and positive finite integrals, the horizontal function centroids of their convolution add

and provided that either or has its function centroid at the origin, the second raw moments do as well

(Bracewell 1965, p. 142), where

For cumulative distribution functions and , their convolution in probability theory is given by

where the Stieltjes integral is taken over the real line.

See also

Autoconvolution Inequality, Autocorrelation, Cauchy Product, Convolution Matrix, Convolution Theorem, Cross-Correlation, Recurrence Plot, Riesz Potential, Wiener-Khinchin Theorem Explore this topic in the MathWorld classroom

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References

Bracewell, R. "Convolution" and "Two-Dimensional Convolution." Ch. 3 in The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 25-50 and 243-244, 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.Weisstein, E. W. "Books about Convolution." http://www.ericweisstein.com/encyclopedias/books/Convolution.html.

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Convolution

Cite this as:

Weisstein, Eric W. "Convolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Convolution.html

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