The paper discusses signals with a finite rate of innovation, which are not necessarily bandlimited but can be uniformly sampled at or above their rate of innovation. The authors derive sampling theorems for periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. They also present local reconstruction schemes for infinite-length signals with a finite local rate of innovation using spline kernels. The key to these constructions is identifying the innovative part of the signal using an annihilating or locator filter, a technique commonly used in spectral analysis and error-correction coding. The paper provides experimental results to support the theoretical findings and discusses potential applications in signal processing, communications systems, and biological systems.The paper discusses signals with a finite rate of innovation, which are not necessarily bandlimited but can be uniformly sampled at or above their rate of innovation. The authors derive sampling theorems for periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. They also present local reconstruction schemes for infinite-length signals with a finite local rate of innovation using spline kernels. The key to these constructions is identifying the innovative part of the signal using an annihilating or locator filter, a technique commonly used in spectral analysis and error-correction coding. The paper provides experimental results to support the theoretical findings and discusses potential applications in signal processing, communications systems, and biological systems.