Periodic summation



In mathematics, any integrable function $$s(t)$$ can be made into a periodic function $$s_P(t)$$ with period P by summing the translations of the function $$s(t)$$ by integer multiples of P. This is called periodic summation:


 * $$s_P(t) = \sum_{n=-\infty}^\infty s(t + nP)$$

When $$s_P(t)$$ is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, $$S(f) \triangleq \mathcal{F}\{s(t)\},$$ at intervals of $$\tfrac{1}{P}$$. That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of $$s(t)$$ at constant intervals (T) is equivalent to a periodic summation of $$S(f),$$ which is known as a discrete-time Fourier transform.

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

Quotient space as domain
If a periodic function is instead represented using the quotient space domain $$\mathbb{R}/(P\mathbb{Z})$$ then one can write:


 * $$\varphi_P : \mathbb{R}/(P\mathbb{Z}) \to \mathbb{R}$$
 * $$\varphi_P(x) = \sum_{\tau\in x} s(\tau) ~ .$$

The arguments of $$\varphi_P$$ are equivalence classes of real numbers that share the same fractional part when divided by $$P$$.