Convolution product

In the space of square-summable functions f: G → R, defined on an abelian group G with values in some ring R, the convolution product

$f*g:={\begin{cases}G\to R\\x\mapsto \int _{y\in G}f(y)\,g(x-y)\,\mathrm {d} y\end{cases}}$

is well defined.

In general, "dy" represents here a Haar measure on that group. If G is a discrete group (like G = Z, the integers), then $\int _{y\in G}\!\!\cdots \mathrm {d} y$ "simplifies" to $\sum _{y\in G}$.

Polynomials

The product of polynomials is a particular case of such a convolution product, given the formula

$c_{n}=\sum _{k=0}^{n}a_{k}\,b_{n-k}$

for the coefficients $c_{n}$ of the product ${\bigl (}\sum a_{k}\,X^{k}{\bigr )}\times {\bigl (}\sum b_{k}\,X^{k}{\bigr )}$. This formula for $c_{n}$ results from the above formula for $a*b$ if the expression $a(k)\,b(n-k)$ is taken to vanish when k or n-k is negative.

Sequences

The same formula can also serve to define the convolution c = a * b of any (not necessarily finitely supported) sequences (a_k) and (b_k).

If R is a k-algebra, then the convolution product has the linearity (or distributive) properties

$(f+\lambda \,g)*h=f*h+\lambda \,g*h~,~~f*(g+\lambda \,h)=f*g+\lambda \,f*h~,~~$

if R is commutative then so is *.

The above linearity properties imply that each element f ∈ R^G defines a k-linear operator ${\tilde {f}}:=(g\mapsto f*g)$ acting on the k-module R^G. This associates to each sequence of OEIS, a transformation acting on each of the other sequences, via convolution.

Examples

(to be written)

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