O'Reilly logo

The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

III.91 Transforms

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, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required