sampling - Absolute convergence of periodic sinc interpolation

An $N$-periodic complex discrete-time sequence $[x_0, \dots, x_{N-1}]$ can be resampled to an $M$-periodic sequence $[y_0, \dots, y_{M-1}]$ with $M>N$, using sinc interpolation:

$$\begin{align}y_m &= \sum_{n=-\infty}^\infty \operatorname{sinc}\left(\frac{Nm}{M} - n\right)x_{n\operatorname{mod}N} \\&= \sum_{k=-\infty}^{\infty}\sum_{n=0}^{N-1}\operatorname{sinc}\left(\frac{Nm}{M} - n - Nk\right)x_n\end{align}\tag{1}$$

where $\operatorname{mod}$ denotes the modulo operation and:

$$\operatorname{sinc}(x) = \begin{cases}1&\text{if }x=0,\\\frac{\sin(\pi x)}{\pi x}&\text{otherwise}.\tag{2}\end{cases}$$

Eq. 1 can be seen as resampling an $N$-periodic continuous-time signal from samples $x_n$ at times $n + Nk$ to samples $y_m$ at times $\frac{Nm}{M}$.

For example, a $2$-periodic complex discrete-time sequence $[x_0, x_1]$ can be resampled to a $4$-periodic sequence $[y_0, y_1, y_2, y_3]$:

$$\text{Eq. 1, }N=2,\,M=4$$ $$\Rightarrow\left\{\begin{align}y_0 &= x_0\\ y_1 &= \sum_{k=-\infty}^\infty\Bigg(\operatorname{sinc}\left(2k+\frac{3}{2}\right)x_0 + \operatorname{sinc}\left(2k+\frac{1}{2}\right)x_1\Bigg)\\ y_2 &= x_1\\ y_3 &= \sum_{k=-\infty}^\infty\Bigg(\operatorname{sinc}\left(2k+\frac{1}{2}\right)x_0 + \operatorname{sinc}\left(2k+\frac{3}{2}\right)x_1\Bigg) \end{align}\right.\tag{3}$$

The two series in Eq. 3 converge conditionally, with for example these possible rearrangements of the first series that give conflicting results if $x_0 \ne x_1$:

$$\begin{gather}\sum_{k=0}^\infty\bigg(f(-k) + f(k+1)\bigg)\\= \frac{x_0 + x_1}{2},\\ \sum_{k=0}^\infty\bigg(f(-k) + f(2k+1) + f(2k+2)\bigg)\\= \frac{(x_1-x_0)\ln(2)}{2\pi} + \frac{x_0 + x_1}{2},\\ \sum_{k=0}^\infty\bigg(f(-k) + f(3k+1) + f(3k+2) + f(3k+3)\bigg)\\= \frac{(x_1-x_0)\ln(3)}{2\pi} + \frac{x_0 + x_1}{2}, \end{gather}\tag{4}$$

with shorthand $f(k) = \operatorname{sinc}\left(2k+\frac{3}{2}\right)x_0 + \operatorname{sinc}\left(2k+\frac{1}{2}\right)x_1$.

Under which condition does the series given by Eq. 1 converge absolutely?