[R] (OT) Fourier coefficients.
Rolf Turner
rolf at math.unb.ca
Tue May 25 21:32:07 CEST 2004
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
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