Hi! I wish to find Fourier coefficients satisfying
(1)
(2)
(3)
(4)
(5)
Here, and are unknown coefficients, and are known coefficients, and is a periodic set in with periodicity , and is defined by eq. (5). The sum in eq. (1) and (4) runs over integers according to eq. (2). My actual problem is two dimensional. Hence, the sums are over and , and the set is defined in two dimensions. I don't know if this changes the problem very much. I'm thankful for any help at all.
So the original two dimensional problem is:
(1)
(2)
(3)
(4)
(5)
The explicit expressions for and are given by:
Every single quantity in the expressions for and are given. is the Heaviside step function. I don't know if these expressions make life much easier. Just to clearify, eqs (1) and (4) just tell that the functions that the Fourier coefficients describe (we can call them and ) are determined to be zero outside and inside the region defined by eq. (5) respectively.
Actually I've found at least one way to solve the problem, and that is to discretize and transform and into one dimensional vectors. This turns all the equations into linear matrix equations that can be solved. However, this only gives an approximate numerical solution that depends on the choice of the grid size in and . An analytical solution for as a function of and would have been a bit nicer. =P