In "The Contourlet Transform: An Efficient Directional Multiresolution Image Representation" by Minh N. Do and Martin Vetterli on page 10 it says:
Definition. A $2D$-function $\lambda(t)$ is said to have an $L$-th order vanishing moment in the direction $u = (u_1, u_2)^T$ if all $1D$-slices of that function along the direction $u$, $\lambda_{u, r_2}(r_1) = \lambda(r_1 u + r_2 u^{\perp})$, where $u^{\perp} = (- u_2, u_1)^T$ have $L$ vanishing moments: $$ \tag{1} \int_{\mathbb{R}} \lambda_{u,r_2}(r_1) r_1^p d r_1 = 0 \quad \forall r_2 \in \mathbb{R}, \ \forall p \in \{0, \ldots, L - 1\}. $$
It can be shown that in the Fourier domain the DVM condition $(1)$ is equivalent to requiring $\Lambda(\omega_1,\omega_2)$ and its $L - 1$ first derivatives in the direction $u$ to be zero along the line $u_1 \omega_1 + u_2 \omega_2 = 0$.
I am interested in a reference for the proof of the statement above or even better a hint on how to prove the following special case:
Problem. Let $\psi: \mathbb{R}^2 \to \mathbb{R}^2$ be compactly supported and for a fixed $\xi_2 \in \mathbb{R}$ fulfill $$\hat{\psi}(0, \xi_2) = \frac{\partial \hat{\psi}}{\partial \xi_1}(0, \xi_2) = 0.$$ Then $\int_{\mathbb{R}} x_1^k \psi(x_1, x_2) d x_1 = 0$ holds for all $x_2 \in \mathbb{R}$ and $k \in \{0,1\}$.
In this problem, $L = 2$, $u = (1,0)^T$, therefore $\psi_{u,r_2}(r_1) = \psi(r_1, r_2)$.