Difference between revisions of "Convolution"

Latest revision as of 18:12, 7 June 2007

The convolution of two functions can mean various things.

In number theoretic context, convolution of two functions $f,g : \mathbb{N} \rightarrow \mathbb{C}$ usually means Dirichlet convolution, defined as $\displaystyle f * g = \sum_{d\mid n} f(d)g\left( \frac{n}{d} \right)$ .

In analytic context, convolution of functions $\displaystyle f, g$ usually means a function of the form $\int f(\tau) g(t-\tau) d\tau$ .

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-For two functions <math> f, g : \mathbb{N} \rightarrow \mathbb{C} </math>, the '''Dirichlet convolution''' (or simply '''convolution''') <math> \displaystyle f * g </math> of <math> \displaystyle f </math> and <math> \displaystyle g </math> is defined as
+The '''convolution''' of two functions can mean various things.
-<center>
-<math> \sum_{d\mid n} f(d)g\left( \frac{n}{d} \right) </math>.
+In [[number theory | number theoretic]] context, convolution of two functions <math> f,g : \mathbb{N} \rightarrow \mathbb{C} </math> usually means [[Dirichlet convolution]], defined as <math> \displaystyle f * g = \sum_{d\mid n} f(d)g\left( \frac{n}{d} \right) </math>.
-</center>
-We usually only consider positive divisors of <math> \displaystyle n </math>.  We are often interested in convolutions of weak [[multiplicative function]]s; the set of weak multiplicative functions is closed under convolution.  In general, convolution is commutative and associative; it also has an identity, the function <math> \displaystyle f(n) </math> defined to be 1 if <math> \displaystyle n=1 </math>, and 0 otherwise.  However, not all functions have inverses (e.g., the function <math> \displaystyle f(n) : n \mapsto 0 </math> has no inverse, as <math> \displaystyle f*g = f </math>, for all functions <math> g: \mathbb{N} \rightarrow \mathbb{C} </math>), although many do.
+In [[analysis | analytic]] context, convolution of functions <math> \displaystyle f, g </math> usually means a function of the form <math> \int f(\tau) g(t-\tau) d\tau </math>.
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Difference between revisions of "Convolution"

Latest revision as of 18:12, 7 June 2007