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In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions ''f'' and ''g'', producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated. Convolution is similar to cross-correlation. It has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations. The convolution can be defined for functions on groups other than Euclidean space. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by ''periodic convolution''. (See row 10 at DTFT#Properties.) A ''discrete convolution'' can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. ==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: :\ \int_^\infty f(\tau)\, g(t - \tau)\, d\tau |- | | (commutativity) |} While the symbol ''t'' is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function ''f''(''τ'') at the moment ''t'' where the weighting is given by ''g''(−''τ'') simply shifted by amount ''t''. As ''t'' changes, the weighting function emphasizes different parts of the input function. For functions ''f'', ''g'' supported on only (i.e., zero for negative arguments), the integration limits can be truncated, resulting in : f(\tau)\, g(t - \tau)\, d\tau \ \ \ \mathrm \ \ f, g : [0, \infty) \to \mathbb |} In this case, the Laplace transform is more appropriate than the Fourier transform below and boundary terms become relevant. For the multi-dimensional formulation of convolution, see Domain of definition (below). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Convolution」の詳細全文を読む スポンサード リンク
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