Periodic summation

with period P by summing the translations of the function

by integer multiples of P. This is called periodic summation:

[1][2] That identity is a form of the Poisson summation formula.

Similarly, a Fourier series whose coefficients are samples of

at constant intervals (T) is equivalent to a periodic summation of

Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.

If a periodic function is instead represented using the quotient space domain

are equivalence classes of real numbers that share the same fractional part when divided by