Editing Convolution (section)

=== Differentiation ===
In the one-variable case,
: <math>\frac{d}{dx}(f * g) = \frac{df}{dx} * g = f * \frac{dg}{dx}</math>

where <math>\frac{d}{dx}</math> is the [[derivative]]. More generally, in the case of functions of several variables, an analogous formula holds with the [[partial derivative]]:
: <math>\frac{\partial}{\partial x_i}(f * g) = \frac{\partial f}{\partial x_i} * g = f * \frac{\partial g}{\partial x_i}.</math>

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of ''f'' and ''g'' is differentiable as many times as ''f'' and ''g'' are in total.

These identities hold for example under the condition that ''f'' and ''g'' are absolutely integrable and at least one of them has an absolutely integrable (L<sup>1</sup>) weak derivative, as a consequence of [[Young's convolution inequality]]. For instance, when ''f'' is continuously differentiable with compact support, and ''g'' is an arbitrary locally integrable function,
: <math>\frac{d}{dx}(f* g) = \frac{df}{dx} * g.</math>

These identities also hold much more broadly in the sense of tempered distributions if one of ''f'' or ''g'' is a 
[[distribution (mathematics)#Convolution versus multiplication|rapidly decreasing tempered distribution]], a 
compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.

In the discrete case, the [[difference operator]] ''D'' ''f''(''n'') = ''f''(''n'' + 1) − ''f''(''n'') satisfies an analogous relationship:
: <math>D(f * g) = (Df) * g = f * (Dg).</math>