Multiplication and division by z is for changing even to odd and vice-versa. On Thu, Dec 18, 2014 at 12:41 PM, Mike Stay <metaweta@gmail.com> wrote:
On Thu, Dec 18, 2014 at 12:25 PM, Daniel Asimov <asimov@msri.org> wrote:
I don't know, but: Why is multiplication or division by z interpreted as rotation? Or what am I missing?
I thought your text meant to describe this:
(Tf)(z) = i (f(z) + f(-z)) / 2 - (f(z) - f(-z)) / 2
No. Assume f is continuously differentiable so it has a Taylor series. This transform swaps neighboring coefficients in the Taylor series and negates one of them.
Split f into even and odd parts fe and fo. Think of f as a 2-d vector |fe| |fo|
Then rotate via the matrix |0 -1| |1 0| which is like i in that applying the matrix twice is minus the identity.
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com