As part of an exploration of the Abel Summation formula (see here), I am looking at an impulse train $T$ made up of $k$ $\operatorname{sinc}$ pulses at intervals $p$ along the $x$ axis:
$$T(x):=\sum_{i=1}^k \operatorname{sinc} \bigl( \pi (x-i p) \bigr)$$
For example, setting $p=3$ and $k=5$ gives
I am seeking tips on how to apply Abel summation to $T$ - or an explanation of why it isn't possible.
The Wikipedia page linked above (under "Variations") gives a summation formula for the indexed sum of some function $\phi$ that is continuously differentiable on real $x \ge 1$. Define a partial sum $A$ by
$$A(t) = \sum_{0 \le n \le t}a_n$$
for a real number $t$ and a sequence of real numbers $a_n$. Set $a_0 = 0$ and $a_n = 1$ for all $n \ge 1$ so that $A(t) = \lfloor t \rfloor$. Then
$$\begin{aligned} \sum_{1 \le n \le x} a_n \phi(n) &= \lfloor x \rfloor \phi(x) - \int_1^x \lfloor u \rfloor \phi'(u) \, du \\&= x \phi(x) - \{x\} \phi(x) - \int_1^x u \phi'(u) \, du + \int_1^x \{u\} \phi'(u) \, du \end{aligned}$$
Here, $\{y \}$ indicates the fractional part of $y$. In order to evaluate the expressions containing the fractional part function, I use a 'proxy' in the form of an explicit formula for the sawtooth function with a limit taken to ensure that it evaluates to zero for integer values of the argument:
$$s(y) := \frac{1}{2} - \underset{\epsilon \to 0^+}{\text{lim}}\frac{\tan ^{-1}(\cot (\pi (y + \epsilon)))}{\pi }$$
So far, so good. But then I hit a wall.
There are so many variables (plus a dummy variable!) knocking around that I am unsure how to define $\phi$ in a way that allows Abel summation to work. I am also losing track of which variations I have tried - although the main one was defining $\phi(i):=\operatorname{sinc} \bigl( \pi (x-i p) \bigr)$ and differentiating and integrating with respect to $I$ (using Mathematica). It didn't work.
So, could someone pleases suggest how I should proceed, or explain why it won't work?