Learning Operators from data is emerging as a dominant paradigm in the application of machine learning to PDEs. Yet, what exactly entails operator learning still eludes a rigorous characterization. We argue that it is not enough for a model to process functions as inputs and outputs to be characterized as an operator learning framework. Instead, it is essential to impose some form of continuous-discrete equivalence to enable the genuine learning of operators, rather than just discrete representations of them. To this end, we adapt tools from harmonic analysis in the form of frame theory to define representation equivalent neural operators (ReNOs), which are constructed to enforce a suitable form of continuous-discrete equivalence. We investigate whether several existing learning frameworks are ReNOs or not. Then, we will present a novel operator learning paradigm, that of convolutional neural operators (CNOs) which are designed to be ReNOs. CNOs are shown to approximate operators, stemming from a large class of PDEs, to desired accuracy. Moreover, we compare CNOs to existing operator learning algorithms in numerical experiments to demonstrate that CNOs are competitive in performance for a variety of PDEs.