Function of moderate growth: history, motivation, and uses I recently came across functions of moderate growth via Are functions of moderate growth a bornological space? and I was wondering, what are some concrete uses or applications of this space?  Where does it appear and why was it introduced historically?
I'm most interested in the continuous variant, discussed in Section 5 of Garrett - Examples of function spaces.
There is no Wikipedia page so I ask here.
 A: Since you've linked to my question, let me give a first answer. The $M$ in the notation $\mathcal{O}_M$ comes from "multiplication". $\mathcal{O}_M$ is the space of functions that can be multiplied with Schwartz functions/distributions to still give Schwartz functions/distributions. Its fourier transform $\mathcal{O}_C'$ is the space of distributions that can be convolved with Schwartz functions/distributions and still produce Schwartz functions/distributions. Once you accept that $\mathcal{S}$ and $\mathcal{S}'$ are useful to study the Fourier transform and its various applications, the subspaces that let you do multiplication and convolution are equally natural.
A nice observation is that $\mathcal{O}_M$ and $\mathcal{O}_C'$ are algebras w.r.t. multiplication and convolution respectively. So they are quite natural operator algebras on $\mathcal{S}$ and $\mathcal{S}'$.
Useful from time to time is the lemma that convolution of $\mathcal{S}'$ and $\mathcal{S}$ gives you a function in $\mathcal{O}_M$.
A: A small complement to Johannes' excellent answer.
If one applies $(-\Delta)^{-\alpha}$ to a function in $\mathscr{S}$, then the result is smooth but the fast decay at infinity is lost. So a convenient space for the codomain of this transform is $\mathscr{O}_{\rm M}$. In other words a (semiregular) kernel like $\frac{1}{|x-y|^{\beta}}$, say with $0<\beta<d$ when working on $\mathbb{R}^d$, lives in $\mathscr{O}_{{\rm M},x}\widehat{\otimes}\mathscr{S}'_{y}$.
With multilinear algebra, one can make some constructions like, starting
from
$$
A\in V\otimes W \otimes X'\otimes Y
$$
and
$$
B\in V'\otimes X\otimes Z\ ,
$$
constructing some new element
$$
A\bullet B\in W\otimes Y\otimes Z
$$
by "contraction of indices" for the dual pairs of vector spaces $V,V'$ and $X,X'$. One of the key points of Schwartz's theory of distributions, is that one can also do this in infinite dimension, provided the spaces used are like $\mathscr{S},\mathscr{S}'$, etc. It also works with $\mathscr{O}_{\rm M}$.
I used that a lot, e.g., in my article "A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion", using a technique I call multiply and conquer for dealing with $\mathscr{O}_{\rm M}$ thanks to its multiplier space characterization.
See also, my MO answer
Can distribution theory be developed Riemann-free?
and in particular the "high tech proof" therein of the last fact mentioned by Johannes: convolution of elements in $\mathscr{S}$ and $\mathscr{S}'$ produces an element of $\mathscr{O}_{\rm M}$. This is an elementary example of the "multiply and conquer" method.
A: In addition to @Abdelmalek Abdesselam's and @Johannes Hahn's good points, I can add a few things from my own context:
One very non-negotiable thing is that Eisenstein series, with a significant role in the spectral theory of automorphic forms, are not in the relevant $L^2$ space, but are only (in an appropriate sense) "of moderate growth".
The simplest explanatory analogue is the case of Fourier transforms on the real line, and the very standard apparatus there (as mentioned by Abdelmalek and Johannes). To make one comparison, we can realize that to apply a tempered distribution $u$ to the Fourier inversion integral $f(x)=\int e^{2\pi ixy}\widehat{f}(y)\,dy$ it is not obvious, and, in fact, not necessarily correct, that $u(f)=\int u(e^{2\pi ixy})\widehat{f}(y)\,dy$. Namely, the exponential is not a Schwartz function.
Nevertheless, the (purely imaginary…) exponentials are bounded, smooth, etc. So, for example, compactly supported distributions can be sensibly applied to them (compatibly with everything else), and can move inside the Fourier inversion integral.
But in many circumstances one wants to have somewhat more general (though still fairly docile) distributions, not just compactly supported ones. Here, already just on the real line (not to mention the automorphic contexts…), it is possible to tell some needless lies. The basis of the possible lies is that (on the real line), continuous, compactly-supported functions are not sup-norm dense in the collection of all bounded continuous functions. (It's not even about measurability….) Thus, the translation action of $\mathbb R$ on bounded continuous functions on it (with sup norm) is not continuous. (E.g., $\sin(x^2)$). This is very bad… and cannot be counted a pathology, but, rather, a misunderstanding of the proper topology.
I think the fundamental point is that the sup-norm closure of continuous, compactly-supported functions is continuous functions going to $0$ at infinity. Thus, the translation action of $\mathbb R$ is continuous on such functions, with sup norm.
That is, rather than looking just at bounded continuous functions on $\mathbb R$, we look at the spaces $V_t$ (with $t$ real…) of continuous functions $f$ such that $\lim_{x\to \infty}\lvert x\rvert^t\cdot \lvert f(x)\rvert=0$. The continuous, compactly-supported functions are dense in each, so the translation action of $\mathbb R$ is (mercifully) continuous on each.
So, the most technically useful, and topologically useful, version of "bounded continuous" is $\bigcap_{t>0} V_t$, which describes a slightly larger space as a (projective) limit….
Hilariously, the colimit over $t$ (an ascending union) does produce the same topology as the naïve version (the latter incorrect on each limitand).
