This work establishes and validates a Grid Graph Signal Processing (G-GSP) framework for estimating the state vector of a radial distribution feeder. One of the key insights from GSP is the generalization of Shannon's sampling theorem for signals defined over the irregular support of a graph, such as the power grid. Using a GSP interpretation of Ohm's law, we show that the system state can be well approximated with relatively few components that correspond to low-pass Graph Fourier Transform (GFT) frequencies. The target application of this theory is the formulation of a three-phase unbalanced Distribution System State Estimation (DSSE) formulation that recovers the GFT approximation of the system state vector from sparse Advanced Metering Infrastructure (AMI) measurements. To ensure convergence of G-GSP for DSSE, the proposed solution relies on a convex relaxation technique. Furthermore, we propose an optimal sensor placement algorithm for AMI measurements. Numerical results demonstrate the efficacy of the proposed method.