Poisson summation formula

The Poisson summation formula equates a suitably regular function’s sum over a lattice with a scaled sum of its Fourier transform over the dual lattice.

Fourier transform: An operation that decomposes a function into frequency components. Transforming the function turns its lattice samples into frequency-space terms.
Lattice (group): A discrete subgroup of Euclidean space generated by a basis of vectors. The formula sums over a lattice and its corresponding dual subgroup.
Jacobi theta function: A family of functions defined by Gaussian-weighted sums over integers. Applying the formula to a Gaussian yields the theta function’s modular transformation.
Simeon Denis Poisson: A French mathematician whose work spanned mechanics, probability, and mathematical physics. The summation formula bears his name and appeared in his work on Fourier analysis.
Dual lattice: The lattice of linear functionals taking integer values on every vector of a given lattice. Its points index the Fourier-transform samples in the formula.
Periodization: The construction of a periodic function by summing translates of a given function. The lattice sum is a periodization whose Fourier coefficients produce the other side.
Gaussian function: A function proportional to the exponential of a negative quadratic form. Its Fourier transform is another Gaussian, making it the formula’s simplest powerful test case.
Joseph Fourier: A French mathematician who developed the use of trigonometric series to study heat conduction. Fourier series provide the central technique behind a standard derivation of the formula.
Dirac comb: A periodic distribution consisting of equally spaced Dirac delta impulses. Fourier transformation maps a lattice comb to a scaled comb on the dual lattice.
Fourier coefficient: A coefficient measuring a function’s contribution at one frequency in a Fourier expansion. Each dual-lattice sample is a Fourier coefficient of the periodized function, up to normalization.

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