Poisson Summation Formula
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High Quality Content by WIKIPEDIA articles! In mathematics, the Poisson summation formula is an equation relating the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. In partial differential equations, the Poisson summation formula provides a rigorous justification for…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, the Poisson summation formula is an equation relating the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. In partial differential equations, the Poisson summation formula provides a rigorous justification for the fundamental solution of the heat equation with absorbing rectangular boundary by the method of images. Here the heat kernel on R2 is known, and that of a rectangle is determined by taking the periodization. The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions (Grafakos 2004).