1. ## Fourier series problem

Hi! I wish to find Fourier coefficients $a_k$ satisfying

$\sum_{k}{a_k e^{i k x}} = 0~\forall x \notin \mathbb{P}$ (1)

$k = \frac{2 \pi m}{X},~m \in \mathbb{Z}$ (2)

$a_k = b_k c_k + d_k$ (3)

$\sum_{k}{b_k e^{i k x}} = 0~\forall x \in \mathbb{P}$ (4)

$\mathbb{P} \equiv \lbrace x : \exists n \in \mathbb{Z} : 0 \leq x + n X \leq L_x \rbrace,~L_x < X$ (5)

Here, $a_k$ and $b_k$ are unknown coefficients, $c_k$ and $d_k$ are known coefficients, and $\mathbb{P}$ is a periodic set in $x$ with periodicity $X$, and is defined by eq. (5). The sum in eq. (1) and (4) runs over integers $m$ according to eq. (2). My actual problem is two dimensional. Hence, the sums are over $k_x$ and $k_y$, and the set $\mathbb{P}$ is defined in two dimensions. I don't know if this changes the problem very much. I'm thankful for any help at all.

2. Originally Posted by sitho
Hi! I wish to find Fourier coefficients $a_k$ satisfying

$\sum_{k}{a_k e^{i k x}} = 0~\forall x \notin \mathbb{P}$ (1)

$k = \frac{2 \pi m}{X},~m \in \mathbb{Z}$ (2)

$a_k = b_k c_k + d_k$ (3)

$\sum_{k}{b_k e^{i k x}} = 0~\forall x \in \mathbb{P}$ (4)

$\mathbb{P} \equiv \lbrace x : \exists n \in \mathbb{Z} : 0 \leq x + n X \leq L_x \rbrace,~L_x < X$ (5)

Here, $a_k$ and $b_k$ are unknown coefficients, $c_k$ and $d_k$ are known coefficients, and $\mathbb{P}$ is a periodic set in $x$ with periodicity $X$, and is defined by eq. (5). The sum in eq. (1) and (4) runs over integers $m$ according to eq. (2). My actual problem is two dimensional. Hence, the sums are over $k_x$ and $k_y$, and the set $\mathbb{P}$ is defined in two dimensions. I don't know if this changes the problem very much. I'm thankful for any help at all.
This might be easier to understand if you posted the original problem.

CB

3. So the original two dimensional problem is:

$\sum_{\boldsymbol{k}}{a_{\boldsymbol{k}} e^{i \boldsymbol{k}\cdot \boldsymbol{x}}} = 0~\forall \boldsymbol{x} \notin \mathbb{P}$ (1)

$\boldsymbol{k} = \left(\frac{2 \pi m_x}{X}, \frac{2 \pi m_y}{Y}\right),~m_x, m_y \in \mathbb{Z}$ (2)

$a_{\boldsymbol{k}} = b_{\boldsymbol{k}} c_{\boldsymbol{k}} + d_{\boldsymbol{k}}$ (3)

$\sum_{\boldsymbol{k}}{b_{\boldsymbol{k}} e^{i \boldsymbol{k}\cdot \boldsymbol{x}}} = 0~\forall \boldsymbol{x} \in \mathbb{P}$ (4)

$\mathbb{P} \equiv \lbrace \boldsymbol{x} : \exists n_x, n_y \in \mathbb{Z} : 0 \leq x + n_x X \leq L_x, 0 \leq y + n_y Y \leq L_y \rbrace,~L_x < X, L_y < Y$ (5)

The explicit expressions for $c_{\boldsymbol{k}}$ and $d_{\boldsymbol{k}}$ are given by:

$c_{\boldsymbol{k}} \equiv {-\frac{\cosh(\kappa[x_{pr} - x_{sr}])}{\kappa\cosh(\kappa[x_{pr} - x_c])\sinh(\kappa[x_c - x_{sr}])}}$

$d_{\boldsymbol{k}} \equiv b_{z 0}(\boldsymbol{k})\frac{e^{-i k_x x_{pr}} + R e^{i k_x x_{pr}}}{\cosh(\kappa[x_{pr} - x_c])} - {}$
${} - \mu_0 j_{a, y}(\boldsymbol{k})\bigg[\theta(x_c - x_a)\frac{\cosh(\kappa[x_{pr} - x_a])}{\cosh(\kappa[x_{pr} - x_c])} - \theta(x_a - x_c)\frac{\sinh(\kappa[x_a - x_{sr}])}{\sinh(\kappa[x_c - x_{sr}])}\bigg]$

$\kappa = \sqrt{k_x^2 + k_y^2 - \left(\frac{\omega}{c}\right)^2}$
$b_{z 0}(\boldsymbol{k}) = \frac{s(\boldsymbol{k})}{(b_1(\boldsymbol{k}) + i k_x)e^{-i k_x x_{pr}} + (b_1(\boldsymbol{k}) - i k_x)e^{i k_x x_{pr}}}$
$s(\boldsymbol{k}) = \mu_0 \kappa \frac{j_{a, y}(\boldsymbol{k})\sinh(\kappa[x_a - x_{sr}])}{\cosh(\kappa[x_{pr} - x_{sr}])}$
$b_1(\boldsymbol{k}) = \kappa \tanh(\kappa[x_{pr} - x_{sr}])$
$j_{a, y}(\boldsymbol{k}) = \left\lbrace
\begin{array}{lcc}
-I_a\frac{e^{-i k_x L_x} - 1}{2 \pi R k_x}\frac{e^{-i k_y L_y} - 1}{2 \pi r k_y} & : & k_x \neq 0,~k_y \neq 0 \\
I_a \frac{i L_x}{2 \pi R}\frac{e^{-i k_y L_y} - 1}{2 \pi r k_y} & : & k_x = 0,~k_y \neq 0
\end{array}\right.$

$j_{a, y}(\boldsymbol{k}) = \left\lbrace
\begin{array}{lcc}
I_a \frac{e^{-i k_x L_x} - 1}{2 \pi R k_x}\frac{i L_y}{2 \pi r} & : & k_x \neq 0,~k_y = 0 \\
I_a \frac{L_x}{2 \pi R}\frac{L_y}{2 \pi r} & : & k_x = 0,~k_y = 0
\end{array}\right.$

Every single quantity in the expressions for $c_{\boldsymbol{k}}$ and $d_{\boldsymbol{k}}$ are given. $\theta(x)$ 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 $a(\boldsymbol{x})$ and $b(\boldsymbol{x})$) 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 $\boldsymbol{x}$ and transform $\boldsymbol{x}$ and $\boldsymbol{k}$ 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 $x$ and $y$. An analytical solution for $a_{\boldsymbol{k}}$ as a function of $c_{\boldsymbol{k}}$ and $d_{\boldsymbol{k}}$ would have been a bit nicer. =P