Systems of Prony type appear in various signal reconstruction problems such as finite rate of innovation, superresolution and Fourier inversion of piecewise smooth functions. We propose a novel approach for solving Prony-type systems, which requires sampling the signal at arithmetic progressions. By keeping the number of equations small and fixed, we demonstrate that such "decimation" can lead to practical improvements in the reconstruction accuracy. As an application, we provide a solution to the so-called Eckhoff's conjecture, which asked for reconstructing jump positions and magnitudes of a piecewise-smooth function from its Fourier coefficients with maximal possible asymptotic accuracy -- thus eliminating the Gibbs phenomenon.