Standard neural networks for imaging tasks are tied to a certain fixed image resolution they were trained on. Neural operators were proposed in Kovachki, Nikola, et al. “Neural operator: Learning maps between function spaces with applications to PDEs” as a way to generalize neural networks to an infinite-dimensional setting. A possible resolution-invariant discretization is provided by Fourier neural operators (FNO), as proposed in Li, Zongyi, et al. “Fourier neural operator for parametric partial differential equations”. In this work, we prove well-definedness, continuity and differentiability of FNOs as operators acting on Lebesgue spaces. Furthermore, we identify the equivalence between standard convolutional neural networks (CNNs) with FNOs on a fixed resolution and highlight their differences in the multi-resolution setting. In particular, we highlight, that an FNO layer can be interpreted as a CNN layer with an additional trigonometric interpolation.