Convolution
Differentiation

In the one-variable case,

where d d x {\displaystyle {\frac {d}{dx}}} is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:

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) 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,

These identities also hold much more broadly in the sense of tempered distributions if one of f or g is a 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:
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