One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\eta(0)=1$, and $\eta$ has some smoothness conditions (e.g. $C^2$ or $C^\infty$...) and some asymptotic decay conditions (e.g. $\eta$ is compactly supported or decays exponentially...).
Coming from a physics background, this seems one of the most natural approaches to take. And barring some small caveats it nicely encapsulates other approaches like Abel summation ($\eta(x) = e^{-x}$) and Cesaro summation ($\eta(x) = (1-x)_+$). But surprisingly, there seems a massive dearth of mathematical literature about this general class of approaches. (In particular, I've skimmed Hardy's Divergent Series and didn't spot anything about it.) In fact, Terry Tao's classic blog post on the subject seems to be almost the sole reference! One obvious question is simply: are there any good discussions of this family of approaches out there?
However, the question I'm most interested in is: are there any nice consistency results about these approaches? For example, is the following statement true (potentially with some conditions on $\eta$ or $a_n$ if necessary):
Claim: If $\eta$ and $\tilde{\eta}$ both lead to well-defined values $S$ and $\tilde{S}$ for the above limit, then $S=\tilde{S}$.
(Or perhaps even "If any $\eta$ gives rise to a well-defined value $S$ for a series, then some particular stronger summation method M also ascribes $S$ to the series" in the same way that Norlund means force a generalization of the Abel sum to exist.)
This is a cross-post from Math Overflow.