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The plenoptic function (POF) provides a powerful conceptual tool for describing a number of problems in image/video processing, vision, and graphics. For example, image based-rendering can be seen as sampling and interpolation of the POF. In such applications, it is important to characterize the bandwidth of the POF. We study a simple but representative model of the scene where bandlimited signals (e.g. texture images) are "painted" on smooth surfaces (e.g. of objects or walls). We show that in general the POF is not bandlimited unless the surfaces are flat. We then provide simple rules to estimate the essential bandwidth of the POF for this model. Our analysis reveals that, in addition to the maximum and minimum depths, the bandwidth of the POF also depend on the maximum surface slope and maximum frequency of painted signals. With a unifying formalism based on multidimensional signal processing, we can verify several key results in POF processing, such as induced filtering in space and depth correction interpolation, and quantify the necessary sampling rates.