Poisson Summation Formula - Applicability

Applicability

Eq.3 holds provided s(t) is a continuous integrable function which satisfies

for some C, δ > 0 and every t (Grafakos 2004; Stein & Weiss 1971). Note that such s(t) is uniformly continuous, this together with the decay assumption on s, show that the series defining s_P converges uniformly to a continuous function. Eq.3 holds in the strong sense that both sides converge uniformly and absolutely to the same limit (Stein & Weiss 1971).

Eq.3 holds in a pointwise sense under the strictly weaker assumption that s has bounded variation and

(Zygmund 1968).

The Fourier series on the right-hand side of Eq.3 is then understood as a (conditionally convergent) limit of symmetric partial sums.

As shown above, Eq.3 holds under the much less restrictive assumption that s(t) is in L1(R), but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of s_P(t) (Zygmund 1968). In this case, one may extend the region where equality holds by considering summability methods such as Cesàro summability. When interpreting convergence in this way Eq.2 holds under the less restrictive conditions that g(x) is integrable and 0 is a point of continuity of g_P(x). However Eq.2 may fail to hold even when both and are integrable and continuous, and the sums converge absolutely (Katznelson 1976).