Wikipedia:Reference desk/Archives/Mathematics/2013 March 9

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March 9

Examples of convolution

I saw the wiki page, but I couldn't find any examples using actual numbers evaluating the formula. Could you give some examples of convolution, please? Mathijs Krijzer (talk) 22:41, 9 March 2013 (UTC)

Quoted content from our article on Convolution

Definition The convolution of f and g is written f∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform: $(f*g)(t)\ \ \,$   ${\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )\,g(t-\tau )\,d\tau$ $=\int _{-\infty }^{\infty }f(t-\tau )\,g(\tau )\,d\tau .$       (commutativity) Domain of definition The convolution of two complex-valued functions on Rd $(f*g)(x)=\int _{\mathbf {R} ^{d}}f(y)g(x-y)\,dy$ is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g. Circular discrete convolution When a function gN is periodic, with period N, then for functions, f, such that f∗gN exists, the convolution is also periodic and identical to: $(f*g_{N})[n]\equiv \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{f}[m+kN]\right)g_{N}[n-m].\,$ Circular convolution Main article: Circular convolution When a function gT is periodic, with period T, then for functions, f, such that f∗gT exists, the convolution is also periodic and identical to: $(f*g_{T})(t)\equiv \int _{t_{0}}^{t_{0}+T}\left[\sum _{k=-\infty }^{\infty }f(\tau +kT)\right]g_{T}(t-\tau )\,d\tau ,$ where to is an arbitrary choice. The summation is called a periodic summation of the function f. Discrete convolution For complex-valued functions f, g defined on the set Z of integers, the discrete convolution of f and g is given by: $(f*g)[n]\ {\stackrel {\mathrm {def} }{=}}\ \sum _{m=-\infty }^{\infty }f[m]\,g[n-m]$ $=\sum _{m=-\infty }^{\infty }f[n-m]\,g[m].$       (commutativity) When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, extended with zeros where necessary to avoid undefined terms; this is known as the Cauchy product of the coefficients of the two polynomials.

A convolution maps 2 functions to a third function, it does not map numbers to anything or anything to numbers, so unless you are going to point wise define a function in terms of numbers, I can't show you anything "using actual numbers". — Preceding unsigned comment added by 123.136.64.14 (talk) 05:39, 12 March 2013 (UTC)