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

= 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)

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)
 * }
 * $$= \int_{-\infty}^\infty f(t-\tau)\, g(\tau)\, d\tau.$$      (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
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)