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Convolution product

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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

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 "simplifies" to .

Polynomials

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

for the coefficients of the product . This formula for results from the above formula for if the expression 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 (ak) and (bk).

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

if R is commutative then so is *.

The above linearity properties imply that each element f ∈ RG defines a k-linear operator acting on the k-module RG. 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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