Is there an algebra for divergent series summation operators?

Question

Let $D$ denote a divergent series and let $C$ denote a convergent series.

Furthermore, let $s :$ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one of these operators:

(the hyperlinks will direct you to the wiki page of the relevant summation method, not the person who invented/discovered it)

Borel summation
Abel summation
Euler summation
Cesàro summation
Lambert summation
Ramanujan summation
Summing the series by means of Analytic continutation
Some Regularization method

I am wondering if there is any meaningful way to answer the following questions (Assuming $D_1 , D_2$ are summable with $s$ ):

What does $s(D_1 + D_2)$ equal? Is it always equal to $s(D_2 + D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?
What does $s(D_1 \cdot D_2)$ equal? Is it always equal to $s(D_2 \cdot D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?
What happens when we add convergent series into the mix? And what if we're summing linear combinations of $n$ convergent and $m$ divergent series?

Do the results differ for different summation methods, listed above?

(This question was migrated from MSE. I also asked a somewhat similar question on MO once.)

I think the conditions you've listed are not the best one to focus on. Wikipedia suggests regularity (works on convergent series), linearity, and stability (tacking on a new initial term has the expected effect); these strike me as sensible conditions. Your (1) is linearity but without scalars, OK. Your (3) is a weird mix -- regularity, but also linearity again (now with scalars). If something works for linear combinations of 2 things, you don't need to separately consider linear combinations of n things. And if it works for convergent series, you don't need to separately consider linear... — Harry Altman, Apr 25 '13 at 21:41
...combinations of them. So instead of (1) and (3), I think it would be better to just focus on linearity and regularity (though perhaps linearity is worth breaking down into additivity and scaling?). As for (2), well, that's not going to work very well even if your summation method is "take the actual sum" because I don't think that will even work for convergent series if the series are only conditionally convergent. (That's assuming we're defining $D_1\cdot D_2$ how I think we are; you didn't provide a definition.) So yeah, I might suggest replacing these conditions with better ones. — Harry Altman, Apr 25 '13 at 21:45
@Harry Altman: I'm sorry, I'm not entirely sure what you mean. I have listed my conditions on top of the question. These conditions that are listed are the most common ones for divergent series in the wikipedia article on divergent series. The actual questions, which you refer to as (1), (2) and (3) (I guess), and I refer to as 1. , 2. and 3. , ask for the actual outcome of some algebraic combinations of divergent series. Do you think I should focus on different conditions for divergent series (as opposed to the ones listed on wikipedia), or do you think I ought to ask different... (cont'd) — Max Muller, Apr 26 '13 at 20:38
Conditions are questions; questions are conditions. I.e. a condition is a question of the form "Is this condition satisfied?". And a question is a condition of the form "The answer to this question is yes." In other words: Your question 1 is, "Is additivity satisfied, and if not, in what ways does it fail?" Question 3 is, "Are regularity and linearity satisfied, and if not, in what way do they fail?" You're asking questions about conditions and so the condition determines the question. My complaint about 1) and 3) is that I think you havn't broken down the conditions you're asking about in... — Harry Altman, Apr 27 '13 at 06:48
...good ways. Separating additivity from scaling may be good, mixing in regularity with additivity and scaling isn't. And asking about a linear combination of $n$ convergent series is the same as asking about a single convergent series. In short, you haven't boiled your questions down, and this makes them annoying to talk about, since one wants to separate out the different issues first. 2) I just think is a bad condition to focus on since it fails so badly even for the trivial "just sum the series" method. I guess that is worth remarking on, though. — Harry Altman, Apr 27 '13 at 07:03
@Harry Altman: O.K. I think I know what you mean now. I will try to ask better, more focussed answers next time. — Max Muller, Apr 27 '13 at 15:46
I think this answer answers the question: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 — Anixx, Oct 10 '19 at 15:35

score 8 · Accepted Answer · edited Apr 13 '17 at 12:58

I'll try to offer as much knowledge as I can on this topic, which might not be that much.

Firstly your linear form $s$ is defined on a strict linear subspace $L$ of the vector spaces of all formal series. Secondly you seem to mix formal numeric series and formal power series, which are distinct rings.

1) 2) By definition of the ring of formal (numeric or power) series over $\mathbb C$ , the addition and product are commutative, so $s(D1+D2)=s(D2+D1)$ and $s(D1\cdot D2)=s(D2\cdot D1)$ . Now if $s$ is $\mathbb C$ -linear then $s(D1+D2)=s(D1)+s(D2)$ . This is not true anymore for the product, even for convergent series as the product of two conditionnally convergent series might only be divergent. If both are absolutely convergent then the Cauchy-product formula shows that $s(D1\cdot D2)=s(D1)s(D2)$ . This formula for convergent series is related to 3) below.

3) Every method you mention agrees with the usual sum for convergent series (except maybe Ramanujan's which I don't know about). So adding finitely many convergent series to the mix does not change anything by linearity. By the way the "summation by analytic continuation" is often refferred to as Mittag-Leffler's summation. Be careful though that it is not well defined due to the fact that most analytic continuations yield multivalued functions.

I think I should mention a nice paper by Lyubich which tries to axiomatically define what should be a coherent summation method for numeric series. The main property is that, following the afore-mentionned axioms, the sum $1+1+1+\cdots$ will never be assigned a finite value, i.e. it does not belong to $L$ .

Also as far as I know the regularization process in QFT first transform a "series" of infinite quantities into a formal power series over $\mathbb C$ which may be (is) divergent, as explained somehow in this thread. In some cases the latter is a Borel-summable power series. See this survey (in French) by J.-P. Ramis for more details regarding this topic.

You might also want to learn more about Borel-summation for Gevrey power series through the works of Ramis, Sibuya, Balser and others and also through the mould/alien calculus developed by Écalle. An example is presented in my answer here.

Thank you. I will check out the papers and sources you mentioned, they seem rather interesting. Regarding 1)2), you say that: "Now if $s$ is $\mathbb{C}$ -linear then $s(D1 + D2) = s(D1) + s(D2)$ ." What I am curious about, is whether divergent series summation operators acutually are $\mathbb{C}$ -linear. Do you know whether this is true? And for which operators? — Max Muller, Apr 26 '13 at 21:17
Well, that's what you claim in the question! But yes, they are, as is stated or clear from the formulas in the links you provide. But again, careful with the product, and with the problem of consistency and well-definedness of some of them, as I already pointed out. — Loïc Teyssier, Apr 27 '13 at 07:14
The article is unaccessibe. Can you please at least tell its title? — Anixx, Sep 04 '17 at 21:56
@Anixx Yes, here is the reference : Lyubich, Yu. I., Axiomatic theory of divergent series and cohomological equations. (English summary) Fund. Math. 198 (2008), no. 3, 263–282. — Loïc Teyssier, Sep 05 '17 at 07:42

Is there an algebra for divergent series summation operators?

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