Editing Poisson summation formula (section)

==Applicability==

{{EquationNote|Eq.2}} holds provided <math>s(x)</math> is a continuous [[Lp space|integrable function]] which satisfies <math display="inline">|s(x)| + |S(x)| \le C (1+|x|)^{-1-\delta}</math> for some <math>C > 0,\delta > 0</math> and every <math>x.</math><ref name="Grafakos"/><ref name="Stein"/> Note that such <math>s(x)</math> is [[uniformly continuous]], this together with the decay assumption on <math>s</math>, show that the series defining <math>s_{_P}</math> converges uniformly to a continuous function. {{EquationNote|Eq.2}} holds in the strong sense that both sides converge uniformly and absolutely to the same limit.<ref name="Stein"/>

{{EquationNote|Eq.2}} holds in a [[pointwise convergence|pointwise]] sense under the strictly weaker assumption that <math>s</math> has bounded variation and<ref name="Zygmund"/>

<math display="block">2 \cdot s(x)=\lim_{\varepsilon\to 0} s(x+\varepsilon) + \lim_{\varepsilon\to 0} s(x-\varepsilon).</math>

The Fourier series on the right-hand side of {{EquationNote|Eq.2}} is then understood as a (conditionally convergent) limit of symmetric partial sums.

As shown above, {{EquationNote|Eq.2}} holds under the much less restrictive assumption that <math>s(x)</math> is in [[Lp space|<math>L^1(\mathbb{R})</math>]], but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of <math>s_{_P}(x).</math><ref name="Zygmund" /> In this case, one may extend the region where equality holds by considering summability methods such as [[Cesàro summation|Cesàro summability]]. When interpreting convergence in this way {{EquationNote|Eq.2}}, case <math>x=0,</math> holds under the less restrictive conditions that <math>s(x)</math> is integrable and 0 is a point of continuity of <math>s_{_P}(x)</math>.  However, {{EquationNote|Eq.2}} may fail to hold even when both <math>s</math> and <math>S</math> are integrable and continuous, and the sums converge absolutely.<ref name="Katznelson" />