We present a new framework for processing point-sampled objects using spectral methods. By establishing a concept of local frequencies on geometry, we introduce a versatile spectral representation that provides a rich repository of signal processing algorithms. Based on an adaptive tesselation of the model surface into regularly resampled displacement fields, our method computes a set of windowed Fourier transforms creating a spectral decomposition of the model. Direct analysis and manipulation of the spectral coefficients supports effective filtering, resampling, power spectrum analysis and local error control. Our algorithms operate directly on points and normals, requiring no vertex connectivity information. They are computationally efficient, robust and amenable to hardware acceleration. We demonstrate the performance of our framework on a selection of example applications including noise removal, enhancement, restoration and subsampling.