Electroencephalography is an important diagnostic tool for functional investigations of the human brain. Recent EEG measurement technologies provide high numbers of electrodes and sampling rates, which results in a considerable quantity of data. For the analysis of this EEG data, efficient signal analysis and decomposition methods are essential. In this paper a new method for spatial harmonic analysis of EEG data using the Laplacian eigenspace of the meshed surface of electrode positions is presented. The resulting eigenspace enables the spatial harmonic analysis, filtering, denoising and decomposition of EEG data. For a proof of concept, the proposed approach is applied to an 128 channel EEG recording of visual evoked potentials. A set of harmonic spatial basis functions for the EEG electrode setup is estimated. The EEG data are spatially decomposed and low pass filtered using the harmonic spatial basis functions.