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T. W. Körner

If we have a finite sequence a0, a1, . . . , an of real numbers (written briefly as a), then we can look at the polynomial

Pa(t)= a0 + a1t + ··· + antn.

Conversely, given a polynomial Q of degree m ≤ n, we can recover a unique sequence b0, b1, . . . , bn such that

Q (t) = b0 + b1t + · · · + bntn

by, for example, taking bk = Q (k) (0) / k!.

We observe that if a0, a1, . . . ,an , and b0, b1, . . . , bn are finite sequences, then

Pa(t)Pb(t) = pa*b(t),

where a * b = c is a sequence c0, cl, . . . , c2n given by

ck = a0bk + albk-l + . . . + akb0,

where we interpret ai, and bi as 0 if i > n. This sequence is called the convolution of the sequences a and b.

To see the kind of use that one can make of this observation, ...

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