Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, piecewise polynomial or piecewise sinusoidal signals, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. In this paper, we show that many signals with finite rate of innovation can be sampled

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... and perfectly reconstructed using kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, Exponential Splines and functions with rational Fourier transforms. Extension of such results to the 2-dimensional case are also discussed and an application to image super-resolution is presented.