Operating power systems with large amounts of renewables requires predicting future photovoltaic (PV) production with fine temporal and spatial resolution. State-of-the-art techniques combine numerical weather predictions with statistical post-processing, but their resolution is too coarse for applications such as local congestion management. In this paper we introduce computing methods for multi-site PV forecasting, which exploit the intuition that PV systems provide a dense network of simple weather stations. These methods rely entirely on production data and address the real-life challenges that come with them, such as noise and gaps. Our approach builds on graph signal processing for signal reconstruction and for forecasting with a linear, spatio-temporal autoregressive (ST-AR) model. It also introduces a data-driven clear-sky production estimation for normalization. The proposed framework was evaluated over one year on both 303 real PV systems under commercial monitoring across Switzerland, and 1000 simulated ones based on high-resolution weather data. The results demonstrate the performance and robustness of the approach: with gaps of four hours on average in the input data, the average daytime NRMSE over a six-hour forecasting horizon (in 15 min steps) and over all systems is 13.8% and 9% for the real and synthetic data sets, respectively.