Related Experiment Video
Updated: Jul 20, 2026

Dissolution Dynamic Nuclear Polarization Instrumentation for Real-time Enzymatic Reaction Rate Measurements by NMR
Published on: February 23, 2016
Dualizing the Poisson summation formula
1Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA.
Abstract:
If f(x) and g(x) are a Fourier cosine transform pair, then the Poisson summation formula can be written as 2sumfrominfinityn = 1g(n) + g(0) = 2sumfrominfinityn = 1f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let phi(x) and gamma(x) = phi(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = sumfrominfinityn = 1phi(n/x)/x - integralinfinity0phi(t)dt and G(x) = sumfrominfinityn = 1gamma (n/x)/x - integralinfinity0 gamma(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of phi because it is the error in estimating the integral phi of by its Riemann sum with the constant mesh spacing 1/x.
Related Concept Videos
Poisson Probability Distribution
The...
Poisson's And Laplace's Equation
Poisson's Ratio
Properties of the z-Transform II
Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency...
Sums of Power
Summation Notation

