Does summing divergent series using cutoff functions give consistent results?

Question

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 (at least continuous at 0, possibly more, e.g. $C^1$ or $C^\infty$) and some asymptotic decay conditions (e.g. $\eta$ is compactly supported or decays exponentially).

Are there any nice consistency results about such 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 convergence of any Norlund mean forces a generalized Abel sum to exist and have that value.)

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Context: 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 ($\eta(x) = (1-x)_+$) and normal summation ($\eta(x) = \mathbb{1}_{[0,1]}(x)$); note these do have such a consistency result. 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! So another obvious question is simply: are there any good discussions of this family of approaches out there?

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Additional Thoughts (31/12/2023): One natural way to think about this problem is to look at the 'simplest' non-trivial divergent $a_n$ possible, which is probably Grandi's series, $a_n = (-1)^{n-1}$. Some observations:

• As mentioned by Tao, if $\eta$ is compactly supported and $C^1$ (in fact I think it suffices to be continuous and piecewise $C^1$), then you necessarily get convergence and it necessarily gives $S = \frac{1}{2}$. (I also suspect you can weaken the compact support condition somewhat but haven't tried to prove this.)
• Obviously with the conventional discontinuous $\eta(x) = \mathbb{1}_{[0,1]}(x)$ this does not converge to anything.
• If you take $\eta(x) = \mathbb{1}_{[0,2]}(x)$, and only consider $N=1,2,3,\ldots$ to take discrete values, then you can obtain $S = 0$. This definitely feels against the spirit of things, but it is an important point to bear in mind.
• Less obviously, I can't see an easy way to construct a discontinuous $\eta$ which leads to convergence to any $S$ if you assume that you take $N$ to vary continuously.

This suggests it's helpful to think about a weaker statement along the lines of (1) all sufficiently smooth $\eta$ agree and a stronger conjecture that (2) all $\eta$ agree if $N$ is taken to vary continuously.

It seems plausible that more familiar arguments like Tao's (based around the Euler-Maclaurin formula) or something involving Mellin transforms can prove a weaker claim along the lines of (1), at least for some restricted class like $a_n = O(n^k)$ for any $k$ (and then if $\eta$ is taken to be e.g. $C^k$ and piecewise $C^{k+1}$ and to decay exponentially you can prove convergence to e.g. the same value as the Abel sum).

The stronger claim (2) then may fail to be true, or perhaps it is possible to disprove that convergence to any $S$ is possible for insufficiently smooth (or perhaps insufficiently rapidly decaying) $\eta$. Or perhaps you can even somehow arrange for convergence but only to a consistent value of $S$.

It would be interesting to see proofs or counterexamples for any of the above statements.

Answer 1

Tao's "smoothed sums" can be seen as a particular class of so-called matrix summation methods in the modern (post-Hardy) literature, see e.g. J. Boos, Classical and Modern Methods in Summability (Oxford University Press, 2001) and references therein. For such methods, there are indeed consistency theorems which I'll detail shortly.

In what follows, let $$\omega = \mathbb{K}^\mathbb{N} = \left\{x=(x_n)=(x_n)_{n\in\mathbb{N}}:\mathbb{N}\rightarrow\mathbb{K}\right\}$$ be the vector space of all $\mathbb{K}$-valued sequences (here $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$), $$c = \left\{x\in\omega\ \middle|\ \lim x\doteq\lim_{n\rightarrow\infty}x_n\text{ exists}\right\}$$ the subspace of all convergent sequences, $$c_s = \left\{x\in\omega\ \middle|\ \Sigma\,x\doteq\left(\sum^n_{k=1}x_k\right)\in c\right\}=\left\{x\in\omega\ \middle|\ \lim\Sigma\,x\doteq\sum^\infty_{n=1}x_n\text{ exists}\right\}$$ the subspace of all summable sequences, and $$l^1 = \left\{x\in\omega\ \middle|\ |x|\doteq\left(|x_n|\right)\in c_s\right\}$$ the subspace of all absolutely summable sequences, so that $l^1\subset c_s\subset c\subset\omega$ (recalling that $x\in c_s$ implies that $\lim x=0$). Put differently, $c$ is the domain of the limit (linear) functional $\lim$ and $c_s=\Sigma^{-1}(c)$, where $\Sigma$ is the partial sum linear map, which may be seen as the infinite matrix $$\Sigma=\left[\begin{array}{cccc} 1 & 0 & 0 & \cdots \\ 1 & 1 & 0 & \ddots \\ 1 & 1 & 1 & \ddots \\ \vdots & \vdots & \ddots & \ddots \end{array}\right]\ .$$ Many (linear) summability techniques are based on functional analysis and see the summability problem as a (continuous) extension problem for the series sum (continuous) linear functional $\lim\Sigma$ as defined e.g. in the Banach space $l^1$ to a larger locally convex vector (sequence) space $l^1\subset X\subset\omega$. Such an extension always exists by the Hahn-Banach theorem, so we want a constructive version of that result in this particular context.

The starting point of a matrix summation method is to replace $\Sigma$ with another infinite matrix (likewise identified with the corresponding linear map) $$A=\left[\begin{array}{cccc} a_{11} & a_{12} & a_{13} & \cdots \\ a_{21} & a_{22} & a_{23} & \ddots \\ a_{31} & a_{32} & a_{33} & \ddots \\ \vdots & \vdots & \ddots & \ddots\end{array}\right]$$ such that:

• $X$ is contained in the domain of $A$, that is, $$X\subset\left\{x\in\omega\ \middle|\ Ax\doteq\left(\lim\Sigma\,(a_{nk}x_k)_{k\in\mathbb{N}}\right)_{n\in\mathbb{N}}\text{ exists}\right\}\ ;$$
• $AX\subset c$, so that $\lim Ax$ exists for all $x\in X$;
• At the very least, $\lim A$ extends $\lim\Sigma$ from $$\varphi=\mathbb{K}^{(\mathbb{N})}=\left\{x\in\omega\ \middle|\ \exists k\in\mathbb{N}:x_n=0\ \forall n\geq k\right\}\subset l^1\ ,$$ the subspace of finitely non-zero sequences, to $X$, that is, $\varphi\subset X$ and $\lim Ax=\lim\Sigma\,x$ for all $x\in\varphi$.

There are additional conditions which are discussed e.g. in Boos's book quoted above, but these will not matter in what follows. As a rule, $A$ is chosen first and then $X$ is determined from $A$ and the above conditions - one usually gets a Fréchet sequence space. In the case of smoothed sums with respect to a given cutoff function $\eta$, the matrix $A$ is given by $a_{nk}=\eta\left(\frac{k}{n}\right)$, $n,k\in\mathbb{N}$.

Within such a framework, a consistency theorem is a statement of the following kind: let matrices $A,B$ as above with corresponding sequence subspaces $X_A,X_B$ playing the role of $X$ for each of these, and let $x\in (X_A\cap X_B)\smallsetminus\varphi$.

Theorem (consistency of $A,B$ on $Y$): Suppose $(\lim A)|_{\varphi\oplus\mathbb{K}x}=(\lim B)|_{\varphi\oplus\mathbb{K}x}$. Then there is a sequence subspace $\varphi\oplus\mathbb{K}x\subsetneqq Y\subset X_A\cap X_B$ such that $(\lim A)|_Y=(\lim B)|_Y$.

Section 2.6 and Chapters 9-10 of Boos's book are entirely dedicated to such results.

As a final remark, I would like to stress that the similarity of the above discussion with renormalization ideas in quantum field theory is no accident: as pointed by Klaus Hepp long ago, perturbative renormalization in quantum field theory (with an infrared cutoff in the interaction) is nothing more than constructively extending the distributions corresponding to Feynman graphs, which are usually only defined in a (usually finite-codimensional) subspace of test functions. To directly quote (see pp. 302 of K. Hepp, Proof of the Bogoliubov-Parasiuk Theorem on Renormalization, Commun. Math. Phys. 2 (1966) 301-326):

The renormalization theory (...) is in this framework a constructive form of the Hahn-Banach theorem.