Gaussian Poisson Summation Duality
Use Gaussian lattice duality to choose the faster-converging summation side.
Free theory articleUse Gaussian lattice duality to choose the faster-converging summation side.
Concept & interpretation
The Poisson summation formula establishes a profound connection between a function and its Fourier transform by summing their values over discrete lattices. For Gaussian functions, this manifests as a remarkable identity linking sums over the integer lattice and its reciprocal.
Experiment
Explore the duality by manipulating the Gaussian Poisson Summation Duality.
- Adjust the Gaussian rate control. Observe how a small rate (e.g., 0.25) makes the direct sum converge slowly (leaving a large omitted tail) but makes the reciprocal sum converge very rapidly.
- Set the Gaussian rate to a large value (e.g., 4). Notice that the situation reverses: the direct sum now converges almost immediately, while the reciprocal sum requires more terms.
- Modify the Truncation limit. See how increasing N reduces the omitted tails on both sides, while the mathematically infinite sum and the identity residual (the difference between the infinite limits computed on both sides) remain robustly accurate.