2d Functions and their 2d Fourier coefficients

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Using the conventions of numpy.fft: Normalization by 1/N on the inverse transform, ordering starts at zero mode, then positive modes in ascending order, then negative modes in ascending order (starting at the "most negative") N is always the number of dof

Function	Fourier Coefficients
Example for $N = 5 \times 5$ on $Ω = [0, 2 π)^{2}$
spatial grid	$x = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ \frac{1}{5} 2 π \end{matrix}] & [\begin{matrix} 0 \\ \frac{2}{5} 2 π \end{matrix}] & [\begin{matrix} 0 \\ \frac{3}{5} 2 π \end{matrix}] & [\begin{matrix} 0 \\ \frac{4}{5} 2 π \end{matrix}] \\ [\begin{matrix} \frac{1}{5} 2 π \\ 0 \end{matrix}] & [\begin{matrix} \frac{1}{5} 2 π \\ \frac{1}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{1}{5} 2 π \\ \frac{2}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{1}{5} 2 π \\ \frac{3}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{1}{5} 2 π \\ \frac{4}{5} 2 π \end{matrix}] \\ [\begin{matrix} \frac{2}{5} 2 π \\ 0 \end{matrix}] & [\begin{matrix} \frac{2}{5} 2 π \\ \frac{1}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{2}{5} 2 π \\ \frac{2}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{2}{5} 2 π \\ \frac{3}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{2}{5} 2 π \\ \frac{4}{5} 2 π \end{matrix}] \\ [\begin{matrix} \frac{3}{5} 2 π \\ 0 \end{matrix}] & [\begin{matrix} \frac{3}{5} 2 π \\ \frac{1}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{3}{5} 2 π \\ \frac{2}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{3}{5} 2 π \\ \frac{3}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{3}{5} 2 π \\ \frac{4}{5} 2 π \end{matrix}] \\ [\begin{matrix} \frac{4}{5} 2 π \\ 0 \end{matrix}] & [\begin{matrix} \frac{4}{5} 2 π \\ \frac{1}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{4}{5} 2 π \\ \frac{2}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{4}{5} 2 π \\ \frac{3}{5} 2 π \end{matrix}] & [\begin{matrix} \frac{4}{5} 2 π \\ \frac{4}{5} 2 π \end{matrix}] \end{matrix}]$
wavenumber grid using "ij" convention	$k = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 1 \end{matrix}] & [\begin{matrix} 0 \\ 2 \end{matrix}] & [\begin{matrix} 0 \\ - 2 \end{matrix}] & [\begin{matrix} 0 \\ - 1 \end{matrix}] \\ [\begin{matrix} 1 \\ 0 \end{matrix}] & [\begin{matrix} 1 \\ 1 \end{matrix}] & [\begin{matrix} 1 \\ 2 \end{matrix}] & [\begin{matrix} 1 \\ - 2 \end{matrix}] & [\begin{matrix} 1 \\ - 1 \end{matrix}] \\ [\begin{matrix} 2 \\ 0 \end{matrix}] & [\begin{matrix} 2 \\ 1 \end{matrix}] & [\begin{matrix} 2 \\ 2 \end{matrix}] & [\begin{matrix} 2 \\ - 2 \end{matrix}] & [\begin{matrix} 2 \\ - 1 \end{matrix}] \\ [\begin{matrix} - 2 \\ 0 \end{matrix}] & [\begin{matrix} - 2 \\ 1 \end{matrix}] & [\begin{matrix} - 2 \\ 2 \end{matrix}] & [\begin{matrix} - 2 \\ - 2 \end{matrix}] & [\begin{matrix} - 2 \\ - 1 \end{matrix}] \\ [\begin{matrix} - 1 \\ 0 \end{matrix}] & [\begin{matrix} - 1 \\ 1 \end{matrix}] & [\begin{matrix} - 1 \\ 2 \end{matrix}] & [\begin{matrix} - 1 \\ - 2 \end{matrix}] & [\begin{matrix} - 1 \\ - 1 \end{matrix}] \end{matrix}]$
$u (x) = a$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} a 5^{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a i$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ a 5^{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a \sin (1 x_{0})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ - a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a \cos (1 x_{0})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a i \sin (1 x_{0})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} - a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a i \cos (1 x_{0})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a \sin (1 x_{1})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ - a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a \cos (1 x_{1})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a i \sin (1 x_{1})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} a \frac{5^{2}}{2} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} - a \frac{5^{2}}{2} \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a i \cos (1 x_{1})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ a \frac{5^{2}}{2} \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
$u (x) = a \sin (1 x_{0}) \sin (1 x_{1})$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} - a \frac{5^{2}}{4} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} a \frac{5^{2}}{4} \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} a \frac{5^{2}}{4} \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} - a \frac{5^{2}}{4} \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$
template $u (x) = \dots$	${\hat{u}}_{h} = [\begin{matrix} [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{matrix}] ⊙ [\begin{matrix} 0 \\ i \end{matrix}]$