R help - May 2004 - (OT) Fourier coefficients.

This posting has nothing to do with R (except maybe that I am using R
very heavily in writing the paper to which the question pertains.)  I
simply wish to draw upon the impressive knowledge and wisdom of the R
community.

Since this question is way off topic, if anybody has the urge to
reply, they should probably email me directly:

			rolf at math.unb.ca

rather than via this list.

My question is essentially about Fourier coefficients:

Suppose

                   pi
                   /
	2*pi*a_k = | f(omega)*exp(-i*k*omega) d omega
                   /
                  -pi

and
                   pi
                   /
	2*pi*b_k = | G(omega)*f(omega)*exp(-i*k*omega) d omega
                   /
                  -pi

(The ``*''-s just mean multiplication here, not convolution; i is
of course sqrt(-1).)

The function f() is positive and symmetric about 0 (it's actually
a spectral density function) and G() is the gain of a nice (ARMA)
filter

                   | p(exp(i*omega) |^2
	G(omega) = | -------------- |
                   | q(exp(i*omega) |

where p() and q() are polynomials (with real coefficients); q() has
no zeroes inside the unit disk.

Suppose that the a_k satisfy an asymptotic condition:
a_k * ln k ---> 0 as k ---> infinity.  (The ``Berman condition''.)

Can I say that the b_k satisfy this condition?  If not, where
would I look for a counter-example?  And could I add some extra
not-too-stringent restrictions on the spectrum f() so that I
***could*** say that the b_k satisfy the Berman condition?

Any hints gratefully received.

					cheers,

						Rolf Turner
						rolf at math.unb.ca