Fourier series
I consider a fully-periodic domain \(\left[ 0, \lx \right) \times \left[ 0, \ly \right)\), where a scalar field \(q \in \mathbb{R}^2\) (velocity, pressure, passive scalar in the above equations) can be expanded using the Fourier series (trigonometric functions as the trial functions):
\[q \left( t, x, y \right)
=
\sum_{\kx^{\prime}}
\sum_{\ky^{\prime}}
\wav{q}{\kx^{\prime} \ky^{\prime}} \left( t \right)
\exp \left( I \kx^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky^{\prime} \frac{2 \pi}{\ly} y \right),\]
where \(\kx^{\prime}, \ky^{\prime} \in \mathbb{Z}\) and take \(\left[ - N / 2, - N / 2 + 1, \cdots, N / 2 - 1 \right]\), where \(N\) is the degree of freedom in the direction.
Note that \(\wav{q}{\kx \ky}\) are complex numbers \(\in \mathbb{C}\).
Hereafter the temporal dependency is dropped for notational simplicity.
Integrating this equation in the whole domain with the trigonometric functions as the weighting function
\[\int_{0}^{\ly}
\int_{0}^{\lx}
q \left( x, y \right)
\exp \left( - I \kx \frac{2 \pi}{\lx} x \right)
\exp \left( - I \ky \frac{2 \pi}{\ly} y \right)
dx
dy\]
yields
\[\begin{split}&
\int_{0}^{\ly}
\int_{0}^{\lx}
\sum_{\kx^{\prime}}
\sum_{\ky^{\prime}}
\wav{q}{\kx^{\prime} \ky^{\prime}}
\exp \left( I \kx^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky^{\prime} \frac{2 \pi}{\ly} y \right)
\exp \left( - I \kx \frac{2 \pi}{\lx} x \right)
\exp \left( - I \ky \frac{2 \pi}{\ly} y \right)
dx
dy \\
&
=
\lx \ly \wav{q}{\kx \ky},\end{split}\]
where the orthogonality is used to reach the final relation.
Spatial derivative
For the spatial derivative of this quantity, I have
\[\begin{split}\der{q}{x}
& =
\der{}{x}
\sum_{\kx^{\prime}}
\sum_{\ky^{\prime}}
\wav{q}{\kx^{\prime} \ky^{\prime}}
\exp \left( I \kx^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky^{\prime} \frac{2 \pi}{\ly} y \right) \\
& =
\sum_{\kx^{\prime}}
\sum_{\ky^{\prime}}
I \kx^{\prime} \frac{2 \pi}{\lx}
\wav{q}{\kx^{\prime} \ky^{\prime}}
\exp \left( I \kx^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky^{\prime} \frac{2 \pi}{\ly} y \right),\end{split}\]
and thus the integral
\[\int_{0}^{\ly}
\int_{0}^{\lx}
\der{q}{x}
\exp \left( - I \kx \frac{2 \pi}{\lx} x \right)
\exp \left( - I \ky \frac{2 \pi}{\ly} y \right)
dx
dy\]
yields
\[\lx \ly I \kx \frac{2 \pi}{\lx} \wav{q}{\kx \ky}.\]
Convolution sum
Integrand:
\[p
q
=
\sum_{\kx_0^{\prime}} \sum_{\ky_0^{\prime}}
\wav{p}{\kx_0^{\prime} \ky_0^{\prime}}
\exp \left( I \kx_0^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky_0^{\prime} \frac{2 \pi}{\ly} y \right)
\sum_{\kx_1^{\prime}} \sum_{\ky_1^{\prime}}
\wav{q}{\kx_1^{\prime} \ky_1^{\prime}}
\exp \left( I \kx_1^{\prime} \frac{2 \pi}{\lx} x \right)
\exp \left( I \ky_1^{\prime} \frac{2 \pi}{\ly} y \right),\]
Integral:
\[\begin{split}&
\int_{0}^{\ly}
\int_{0}^{\lx}
pq
\exp \left( - I \kx \frac{2 \pi}{\lx} x \right)
\exp \left( - I \ky \frac{2 \pi}{\ly} y \right)
dx
dy \\
=
&
\int_{0}^{\ly}
\int_{0}^{\lx}
\sum_{\kx_0^{\prime}}
\sum_{\ky_0^{\prime}}
\sum_{\kx_1^{\prime}}
\sum_{\ky_1^{\prime}}
\wav{p}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{q}{\kx_1^{\prime} \ky_1^{\prime}}
\exp \left\{ I \left( \kx_0^{\prime} + \kx_1^{\prime} - \kx \right) \frac{2 \pi}{\lx} x \right\}
\exp \left\{ I \left( \ky_0^{\prime} + \ky_1^{\prime} - \ky \right) \frac{2 \pi}{\ly} y \right\}
dx
dy \\
=
&
\lx \ly
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{p}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{q}{\kx_1^{\prime} \ky_1^{\prime}}.\end{split}\]