Analysis and reconstruction filters are crucial in graphics. The signal processing community has recently developed new filtering strategies based on a generalization of the traditional sampling pipeline. The main idea is to select simple basis functions such as B-splines but to effectively reshape these kernels by adding a discrete transformation filter. This approach is not widely known to graphics practitioners. In this paper we introduce new notation to succinctly summarize important algorithms in generalized sampling. We also present and analyze novel algorithms, including supersampling for antialiased rendering, and image downscaling for mipmap creation. The advantages of generalized sampling are twofold. The non-negativity of B-spline kernels simplifies both importance-based integration and GPU evaluation. And, the broader support of the transformed kernels improves filtering quality. A key challenge is that the discrete transformation often involves inverse convolutions, but fortunately the associated linear systems are banded and can be parallelized.