Briefly: If we modify the hypotheses of Dirichlet's test to require a unimodal sequence of coefficients, not necessarily a monotonic sequence, then do we still get the same quantitative bound on $\sum a_nb_n$? And is this a known result?
Let $(a_n)$ be a sequence of non-negative real numbers that converges to $0$, and let $(b_n)$ be a sequence of complex numbers with bounded partial sums: there exists a constant $B$ such that for all $p\leq q$, we have $\left|\sum_{n=p}^q b_n\right| \leq B$.
If our sequences are supported on the natural numbers, $n\in\mathbb N$, and $(a_n)$ is monotonic, then Dirichlet's test states that $\sum a_nb_n$ converges. The proof proceeds by summation by parts, and in fact, we get $\left|\sum a_nb_n\right|\leq a_0B$.
But what if we generalize to sequences supported on the integers, $n\in\mathbb Z$? Then we should ask for $(a_n)$ to be a unimodal sequence, not a monotonic sequence. That is, $(a_n)$ increases up to some maximum value $a^*$ and then decreases thereafter. So let's do that: $(a_n)$ is now unimodal. But to simplify matters, we'll keep indexing on the natural numbers, $n\in\mathbb N$, anyway, so that convergence is easy to talk about.
Now $(a_n)$ is eventually decreasing, and Dirichlet's test still says $\sum a_nb_n$ converges. What's more interesting is the bound that we can get. The easy approach would be to split the series $\sum a_nb_n$ "horizontally" into two series of $a_nb_n$ terms, one with $a_n$ increasing and the other with $a_n$ decreasing. Then we would get $\left|\sum a_nb_n\right|\leq 2a^*B$. But I believe we can decompose the sum "vertically" into a stack of rectangles, and that should give us the better bound $\left|\sum a_nb_n\right|\leq a^*B$, i.e. without the extra factor of $2$.
Now, I haven't written out a formal proof of the above claim, which might make it hard to evaluate... but I would like to know:
- Did I make a mistake?
- Is this a known result? Is there some reference that I can cite for the inequality $\left|\sum a_nb_n\right|\leq a^*B$?
For comparison, here are some related questions on the site:
- For Dirichlet's test, is it true that if $\ a_1=1\ $ then $\ \left\vert \sum_{n=1}^{\infty} a_n b_n \right\vert \leq M\ ?$ shows the traditional proof for the bound, in the special case of a monotonic sequence of coefficients.
- Is this generalization of Dirichlet's test true? is an attempted generalization that goes too far; if we drop all ordering hypotheses, then the resulting series might diverge.
No amount of Googling has turned up a reference to the claim, so far. (Incidentally, these course notes contain a claim of the usual Dirichlet's test with an extra factor of $2$, but that appears to be unrelated to my question; it's merely due to a typo where $k$ and $k+1$ are reversed, causing a series to telescope incorrectly.)