Question: Product of two Fourier series expansions - Laurent's condition

Hi,

I am trying to get maple to "simplify" the product of two infinite sums, according to Laurent's condition, as follows:

Given two Fourier series expansions,

F = sum{h=-inf..inf, f(h)*exp(i*h*x)}

G = sum{s=-inf..inf, g(s)*exp(i*s*x)}

Their product may be written as F*G, and then simplified as follows:

F*G

= sum{h=-inf..inf, f(h)*exp(i*h*x)} * sum{s=-inf..inf, g(s)*exp(i*s*x)}

= sum{h=-inf..inf, sum{s=-inf..inf, g(s)*exp(i*s*x) * f(h)*exp(i*h*x)}}

= sum{h=-inf..inf, sum{s=-inf..inf, g(s)*f(h)*exp(i*(s+h)*x)}}

define q=h+s, then

= sum{q-s=-inf..inf, sum{s=-inf..inf, g(s)*f(q-s)*exp(i*q*x)}}

or simply

= sum{q=-inf..inf, sum{s=-inf..inf, g(s)*f(q-s)*exp(i*q*x)}}

I'm trying to have Maple do that simplification of changing the summation indices and taking into account the infinite sum, but I don't know how. Any help is appreciated.

Thanks,

 Roy