FPGA Central

The function is:

g(t)=sum(g(n, r)*exp(i * t * (r - n)), n from 0 to +inf),

where r is a negative non-integer, g(n, r) is a function of n and r.

My question is if this g(t) can be expanded into the following Fourier
Series form:

g1(t)=sum(f(n)*exp(i * t * n), n from 0 to +inf),

where f(n) is a function of n, and it is the Fourier Coefficients.

That's to say, can g(t) be expanded into the form of g1(t)?

My difficulty was that the function g(t) looks like not periodic in "t" at
all.

But it should be, because I derived it from a function that was periodic in
"t".

That's to say, be doing some algebra, I hide the periodicity of the function
in "t" and now want to expand it into Fourier Series... is this possible?

Thanks a lot!

Michael wrote:
> The function is:
>
> g(t)=sum(g(n, r)*exp(i * t * (r - n)), n from 0 to +inf),
>
> where r is a negative non-integer, g(n, r) is a function of n and r.
>
> My question is if this g(t) can be expanded into the following Fourier
> Series form:
>
> g1(t)=sum(f(n)*exp(i * t * n), n from 0 to +inf),
>
> where f(n) is a function of n, and it is the Fourier Coefficients.
>
> That's to say, can g(t) be expanded into the form of g1(t)?
>
> My difficulty was that the function g(t) looks like not periodic in "t" at
> all.
>
> But it should be, because I derived it from a function that was periodic in
> "t".
>
> That's to say, be doing some algebra, I hide the periodicity of the function
> in "t" and now want to expand it into Fourier Series... is this possible?

I'm not sure exactly what you mean... exp(i*t) is 2pi periodic. You may
recall that:
exp(i*t) = cos(t) + isin(t)

As long as r is rational, I think g(t) will have periodicity... Or have
I completely missed the subtlety of your question?

A.

In article <eapivt$qq2$[email protected]>,
Michael <[email protected]> wrote:
>The function is:
>
>g(t)=sum(g(n, r)*exp(i * t * (r - n)), n from 0 to +inf),
>
>where r is a negative non-integer, g(n, r) is a function of n and r.
>
>My question is if this g(t) can be expanded into the following Fourier
>Series form:
>
>g1(t)=sum(f(n)*exp(i * t * n), n from 0 to +inf),
>
>where f(n) is a function of n, and it is the Fourier Coefficients.

f(n) = g(r-n, n) would do it, but either the sum for g(t) or
the sum for g1(t) would need n to go from -infty to r. In
general, fourier series have sums from -infty to infty.

Robert Israel [email protected]
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada