## July 17, 2019

### What is the Laplace Transform?

#### Posted by Mike Shulman One of the best ways to understand something difficult is to reinvent it. Two weeks ago I did that with the Laplace transform, and now I feel like it finally makes sense to me. (In fact this was while I was at the Magnitude Workshop, trying to make sense of magnitude for infinite metric spaces. Thanks to Richard Hepworth for pointing out that what I was reinventing was the Laplace transform — in fact I was stumbling towards some of the same ideas that he had already formulated, which are described in his excellent talk.)

The short answer is that the Laplace transform is really just a generalization of the familiar Laurent series representation of complex analytic functions, but where the exponents are allowed to be non-integers and to “vary continuously” rather than discretely. Wikipedia sketches this briefly, but I would never have discovered it there because I was looking for that generalization rather than looking for a way to understand the Laplace transform. Moreover, this explanation is obscured by the fact that people generally choose obfuscating coordinates.

In this post I’ll try to explain the Laplace transform as I understand it now — which is probably still quite rudimentary compared to the people who really understand it, but maybe it’ll be helpful for other folks in the audience who think more like me than like an analyst. (And maybe some analysts will come along and offer further insight!) Along the way we’ll also learn what the “Z-transform” is and obtain some insight into the Fourier transform.

• For any power series $\sum_{n=0}^\infty a_n z^n$ with complex coefficients, if $R = (\limsup_{n\to\infty} |a_n|^{1/n})^{-1}$ then the series converges to a complex-analytic function in the open disc of radius $R$ about the origin.
• Conversely, if $f$ is complex-analytic in the open disc of radius $R\gt 0$ about the origin, then it is equal to a unique power series there.

The following generalization of this is also standard, though it doesn’t always make it into first complex analysis courses:

• For any Laurent series $\sum_{n=-\infty}^\infty a_n z^n$ with complex coefficients, if $R = (\limsup_{n\to\infty} |a_n|^{1/n})^{-1}$ and $r = \limsup_{n\to \infty} |a_{-n}|^{1/n}$ then the series converges to a complex-analytic function in the open annulus of inner radius $r$ and outer radius $R$ centered at the origin.
• Conversely, if $f$ is complex-analytic in a nonempty open annulus centered at the origin, then it is equal to a unique Laurent series there.

Thus, there is a sort of bijection between Laurent series $\sum_{n=-\infty}^\infty a_n z^n$, by which formally I mean just functions $\mathbb{Z} \to \mathbb{C}$, and analytic functions defined on annuli centered at $0$. I say a “sort of” bijection because it may happen that $r=R$, in which case the corresponding open annulus is empty and thus the Laurent series can’t be recovered from any analytic function.

(Note that these Laurent series are infinite in both directions. In particular, this means that two such Laurent series can’t necessarily be multiplied to obtain a new one. For this reason the field of formal Laurent series requires $a_n\neq 0$ for only finitely many $n\lt 0$; analytically this means restricting to functions $f$ analytic in a punctured disc with a pole at the origin. Arbitrary bi-infinite Laurent series also allow essential singularities at the origin, as well as the case $r\gt 0$ of an annulus rather than a punctured disc.)

I recently learned that this operation taking a Laurent series (i.e. a function $a:\mathbb{Z}\to\mathbb{C}$) to its corresponding analytic function $f:\mathbb{C}\to \mathbb{C}$ is known as the Z-transform — except that for some reason people stick in a minus sign, writing $\mathcal{Z}(a_n) = \sum_{n=-\infty}^\infty a_n z^{-n}$. (Wikipedia says that the more mathematically natural convention $\sum_{n=-\infty}^\infty a_n z^n$ is used in geophysics.)

Note that the coefficients of the Laurent series corresponding to an analytic function can be obtained from Cauchy’s integral formula, also known (modulo a minus sign) as the “inverse Z-transform”: if $f(z) = \sum_{n=-\infty}^\infty a_n z^n$ then

$a_n = \frac{1}{2\pi i} \oint f(z) z^{-n-1} \, dz$

where the integral is around a counterclockwise circle contained in the annulus of definition. Now suppose we require the unit circle to be contained in the annulus, i.e. that $r\lt 1\lt R$, which on the Laurent series side amounts to imposing bounds on how fast $a_n$ can grow as $n\to\pm\infty$. If we additionally parametrize the unit circle by $t\in [0,1]$ as $z = e^{2\pi i t}$, with $g(t) = f(e^{2\pi i t})$, then the Laurent series expression on the unit circle becomes

$g(t) = \sum_{n=-\infty}^\infty a_n e^{2 \pi i t n}$

while Cauchy’s integral formula becomes

$a_n = \frac{1}{2\pi i} \int_{t=0}^1 g(t) e^{- 2\pi i t n} \, dt.$

(I’m sure I’m going to mess up the constants somewhere. When I do, please correct me in the comments and I’ll fix it in the post.)

Thus we have sort of recovered the discrete-time Fourier transform, relating functions $\mathbb{Z}\to \mathbb{C}$ with functions $S^1 \to \mathbb{C}$. Of course the general Fourier transform involves functions $S^1 \to \mathbb{C}$ that may not extend analytically to any annulus, and there is a lot more subtlety there. But I like seeing it as arising naturally from Laurent series expansions of analytic functions.

Now, what if we want to consider “power series” with non-integral exponents? This may seem like a weird thing to do, but there are various contexts in which it matters. The one that drew me into it was thinking about magnitude of metric spaces, which naturally involves inverting a matrix whose entries are of the form $q^{d(x,y)}$ for some $q$. The standard choice of $q$ is $e^{-1}$, but then people also scale the metric space by a variable $t$, which amounts to also looking at $q = e^{-t}$. So it seems natural to just treat $q$ as a formal variable, working over some kind of formal polynomial-like ring, and only plug in numbers for it afterwards.

If all the distances $d(x,y)$ in the metric space are integers (e.g. if it is a graph with the shortest-path metric), then $q^{d(x,y)}$ actually lives in a polynomial ring; but otherwise, it has to live somewhere more general. There is a ring $\mathbb{C}[q^{[0,\infty)}]$ of “generalized polynomials”, i.e. finite sums $\sum_{\beta\in S} a_\beta q^\beta$ where $S$ is a finite set of nonnegative real numbers, and it has a fraction field $\mathbb{C}(q^{[0,\infty)})$ of “generalized rational functions”. And for purposes of comparing magnitude to magnitude homology, we want to “do long division” to these generalized rational functions and obtain some kind of power series $\sum_{\beta\in S} a_\beta q^\beta$ where $S\subseteq \mathbb{R}$ can be infinite.

If we want these things to form a field, we have to impose some restrictions on $S$. If we require $S$ to be well-ordered, we get the Hahn series field $\mathbb{C}((q^{\mathbb{R}}))$. This is bigger than necessary for the purpose of magnitude, since the series corresponding to generalized rational functions always have order type $\omega$. However, arbitrary Hahn series of order type $\omega$ aren’t closed under multiplication; e.g. the product of $\sum_{n\in \mathbb{N}} q^n$ by $\sum_{n\in \mathbb{N}} q^{1-1/n}$ has order type $\omega^2$. But instead of expanding to the entire field of Hahn series, we can also restrict to those Hahn series of order type $\omega$ whose exponents approach $\infty$, or equivalently have only finitely many exponents in any interval $(-\infty,N)$ (thereby excluding $\sum_{n\in \mathbb{N}} q^{1-1/n}$). I learned recently that this is a sort of Novikov field, but I don’t know a good notation for it.

Anyway, today we’re not talking about magnitude, and we don’t need our formal series to form a field, but we do want to “evaluate” them at complex numbers and ask whether they converge. It seems to me that a natural restriction for this purpose, analogous to bi-infinite Laurent series $\sum_{n=-\infty}^\infty a_n z^n$, is to require that there are only finitely many exponents in any finite interval $[-N,N]$; but there are certainly other choices one could make. Specifically, for any series $\sum_{\beta\in S} a_\beta q^\beta$ with this property we can evaluate the terms at any given $q$, write down partial sums (either treating both directions $\pm\infty$ separately, or with a “conditional convergence” version that goes in both directions at once), and ask whether they converge.

However… what sort of thing can $q$ be? By analogy with Laurent series, we’d like it to be a complex number; but for a non-integer $\beta$, the function $q^\beta$ of a complex $q$ is multi-valued. Specifically, we can define it as $q^\beta = e^{\beta \log q}$, so every value of $\log q$ gives us a different value of $q^\beta$. Since the infinitely many values of $\log q$ differ by integer multiples of $2\pi i$, the infinitely many values of $q^\beta$ are related by factors of $e^{2\pi i\beta}$ — which is $1$ if $\beta$ is an integer, so that in that case $q^\beta$ is uniquely defined, but not in general otherwise. (If $\beta$ is rational, then $e^{2\pi i\beta}$ is a root of unity, so that $q^\beta$ has finitely many distinct values; but for irrational $\beta$ it has infinitely many.)

The simplest category-theoretically principled way to make sense of a “multi-valued function” $X\to Y$ is to consider it to be an ordinary function on a larger space, i.e. a span $X \leftarrow H \to Y$ such that $H\to X$ is surjective. In the case of analytic functions like $\log q$ and $q^\beta$, we have $X=Y=\mathbb{C}$, while $H$ is a Riemann surface. Since $q^\beta$ is defined in terms of $\log q$, for our purposes it suffices to fix $H$ as the Riemann surface of $\log$, which is a sort of helix fibred over $\mathbb{C}$ with fiber $\mathbb{Z}$ over every nonzero point: I find it pleasing to simply use the variable $q$ to denote a point of this surface $H$. Thus $q$ is not, properly speaking, a complex number itself; but we can take its logarithm (as long as it’s nonzero), and raise it to real exponents, getting a complex number as an answer. If we write $\log q = x+ y i$, then $e^x$ is the absolute value $|q|$, while $y$ is the argument/angle of $q$: note that since $q\in H$ rather than $\mathbb{C}$, its argument is uniquely defined (rather than up to $2\pi$).

Now we can interpret a sum like $\sum_{\beta\in S} a_\beta q^\beta$ as making sense (and possibly converging) for any $q\in H$. I expect that there is a suitable definition of $r$ and $R$ in terms of $a_\beta$ such that this sum converges on the “helical strip” $r\lt |q|\lt R$ to a function analytic on $H$ (maybe someone in the comments will confirm or refute that). However, recovering the series from the function is a thornier problem, since if we “go around a circle” once in $H$ we don’t get back to where we started, so we can’t integrate around a loop as in the Cauchy integral formula.

Before tackling that problem, let’s consider a further generalization. A Laurent series allows us to add up terms $a_n z^n$ for all integers $n$; can we generalize our Novikov series to add up terms $a_\beta q^\beta$ for all real numbers $\beta$ rather than just a countable family of them? (For instance, we might be interested in magnitude of infinite metric spaces.) For this to make sense (and in particular, for it to have a chance of being finite), we ought to take account of the topology of $\mathbb{R}$ and do an integral rather than a sum; thus we’re thinking about “formal integrals” of the form

$\int_{-\infty}^\infty \alpha(\beta)\, q^\beta \,d\beta.$

Our coefficient function $a:\mathbb{Z}\to \mathbb{C}$ has now, of course, been replaced by a function $\alpha:\mathbb{R}\to \mathbb{C}$. But actually, it’s even better to allow $\alpha$ to be a distribution. For one thing, this is the only way the continous version can include the discrete version: given $a:\mathbb{Z}\to \mathbb{C}$, we can take $\alpha = \sum_{n\in \mathbb{Z}} a_n \delta_n$ to be a modulated Dirac comb distribution to make the integral reduce to the sum.

Thus, we can take a complex-valued distribution on $\mathbb{R}$, representing the “coefficients” of a “continuous Novikov series”, and integrate it against $q^\beta$ to obtain an analytic function on some region in $H$ (wherever the integral converges), probably a helical strip $r\lt |q|\lt R$.

This is the Laplace transform!

Unfortunately, people don’t usually write the Laplace transform in terms of $H$ and $q$. Instead, they fix a particular coordinatization of $H$ and work entirely with those coordinates, which (in my view) obscures this nice geometric/analytic picture. To obtain this coordinatization, note that $\log : H \setminus \{0\} \to \mathbb{C}$ is in fact an isomorphism. The inverse of a logarithm is, of course, an exponential map; this means that for $t\in \mathbb{C}$ we can regard $e^t$ as being, not a complex number, but a point of $H$ (the argument of this point “remembers” the whole imaginary part of $t$, as opposed to the complex $e^t\in \mathbb{C}$ which only remembers $Im(t)$ modulo $2\pi$).

If we now stick in another random minus sign, we obtain the coordinatization $(t\mapsto e^{-t}) : \mathbb{C} \to H \setminus \{0\}$ that people generally use. Writing our integral in terms of $q = e^{-t}$, we obtain

$F(t) = \int_{-\infty}^\infty \alpha(\beta)\, e^{-\beta t} \,d\beta.$

which is the standard formula for the two-sided Laplace transform. (The “one-sided” version just restricts $\alpha$ to have support in $[0,\infty)$, giving a sort of continuous analogue of power series rather than Laurent series.)

Note that under this coordinatization, $q\to 0$ corresponds to $Re(t)\to\infty$. Thus, if the $q$-integral converges in a helical strip $r\lt |q|\lt R$ on $H$, the corresponding $t$-integral converges in a vertical strip $-\log(R) \lt Re(t) \lt -\log(r)$ on $\mathbb{C}$. (I say “if” because I expect that it always does, but I haven’t been able to find references on the Internet about the convergence properties of the Laplace transform when $\alpha$ is only a distribution, and since this is just a lark for me I haven’t actually tried to look it up in textbooks.)

Remember that we got the discrete-time Fourier transform from Laurent series by looking at the unit circle? The corresponding thing to look at in this case is the “unit helix” $|q|=1$ in $H$, which corresponds to the imaginary axis $Re(t)=0$. And indeed, if we assume these are contained in the region of convergence, we obtain upon restriction to the line $Re(t)=0$ the standard continuous-time Fourier transform of $\alpha$.

The fiddliest part of the picture is the inverse Laplace transform, and I’m not going to give any details here (partly because I don’t fully understand them myself). Instead of integrating around a circle contained in the annulus of convergence, we integrate along a helix in $H$ contained in the helical strip of $q$-convergence, or equivalently a vertical line contained in the vertical strip of $t$-convergence. In the discrete case, we used the fact that the integrals of $z^n$ around a loop are equal to zero except when $z=-1$, in order to build a sort of Kronecker-delta and extract the coefficients $a_n$ of a Laurent series one by one. In the continuous case, I believe that the integral of $q^\beta$ along the helix actually doesn’t exist for any $\beta$, but it fails to exist in different ways: for $\beta\neq -1$ it oscillates, while for $\beta=-1$ it diverges. Thus, if we are careful about how we pass to the limit in computing the integrals (the formulas I’ve seen appear to be a version of Cauchy principal value), we can use them to build a sort of Dirac delta and extract the “coefficients” function $\alpha(\beta)$.

This method allows us to recover $\alpha$ from its Laplace transform, so that the transform is injective. But characterizing the image of the Laplace transform seems to be much harder than for the Z-transform. In the discrete case, as I recalled above, every analytic function in an annulus is equal to a convergent Laurent series in that annulus. But it seems that the same is not true for analytic functions on helical strips in $H$ — or if it is, I haven’t found anyone pointing it out. (If it were true, I expect it would require the generality of $\alpha$ being a distribution, or at least a measure.) Maybe someone will drop by and give an answer in the comments!

Posted at July 17, 2019 12:45 AM UTC

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### Re: What is the Laplace Transform?

I’ve been wanting to read this post properly since the day it went up, but I’ve been so exhausted from CT2019 (and, I suppose, the magnitude workshop) that I haven’t managed. But let me try one question.

Near the start, you wrote:

The short answer is that the Laplace transform is really just a generalization of the familiar Laurent series representation of complex analytic functions, but where the exponents are allowed to be non-integers and to “vary continuously” rather than discretely

This immediately made me think of the Fourier transform as a kind of generalization of Fourier series: you could replace both “Laplace” and “Laurent” by “Fourier” in your sentence above, and it would still more or less be true. And indeed, you discussed this comparison a couple of times later.

But what we have in Fourier-land is the excellent general theory of Fourier transforms on locally compact abelian groups. As you well know, both the Fourier transform on $\mathbb{R}^n$ and Fourier series of periodic functions are special cases of this general theory.

Specifically, given a suitable function $f$ on $\mathbb{R}^n$, its Fourier transform $\hat{f}$ is the function on $\mathbb{R}^n$ given by

$\hat{f}(\xi) = \int_{\mathbb{R}^n} e^{-2\pi i \langle \xi, x \rangle} f(x)\, dx,$

and given a suitable function $f$ on the unit circle $\mathbb{T}$, its Fourier “transform” $\hat{f}$ is the function on $\mathbb{Z}$ given by

$\hat{f}(k) = \int_{\mathbb{T}} e^{-2\pi i k x} f(x) \, dx.$

In both cases, the domain of $\hat{f}$ is the group of characters $\mathbf{TopGp}(G, \mathbb{T})$ of the domain $G$ of $f$. In the second case, I put “transform” in quotes because the double sequence $\hat{f} = (\hat{f}(k))_{k \in \mathbb{Z}}$ is normally represented as a series $\sum_{k = -\infty}^\infty \hat{f}(k) e^{2\pi i k x}$ and called the Fourier series of $f$, but that’s only a wrinkle of mathematical language.

My question is: is there an analogous overarching framework for the Laplace/Laurent situation? Or if we want an overarching framework, is the best option simply to translate everything into Fourier terms?

Posted by: Tom Leinster on July 19, 2019 1:25 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Good question! I didn’t know the answer, but Google leads me to a 1948 paper by George Mackey with the suggestive title The Laplace transform for locally compact abelian groups, which begins:

…the classical theory of the Fourier transform… has a natural generalization to a theory of a transform taking functions defined on a locally compact Abelian group $G$ into functions defined on the character group $\hat{G}$ of $G$

It is the purpose of the present note to describe how this idea may be further exploited so as to generalize the Laplace transform and the Laurent series in an analogous fashion and thus to connect certain aspects of complex variable analysis with the theory of topological groups in the same way that this is done for real variable analysis by the generalized Fourier transform.

At the present time we are in a position only to describe the basic notions and state a few preliminary results. We expect to publish later in another journal a paper giving detailed proofs of the theorems stated here as well as those of others which we hope to obtain.

A minute of searching didn’t turn up the promised followup paper. But it did turn up a mention of this paper in a post here on the Cafe almost a decade ago!

Posted by: Mike Shulman on July 19, 2019 9:51 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Shooting from the hip:

Fourier transforms generalize nicely to Pontryagin duality for locally compact Hausdorff abelian topological groups. Of these, the nicest are the abelian Lie groups. I think any such thing, say $A$, has a complexification $\mathbb{C}A$, which is a complex Lie group, and an embedding $A \hookrightarrow \mathbb{C}A$. The examples that matter the most are $\mathbb{R} \hookrightarrow \mathbb{C}$ and $\mathrm{S}^1 \hookrightarrow \mathbb{C}^*$. In general, I think the Fourier transform on $A$ should analytically continue to some sort of Laplace transform on $\mathbb{C}A$.

Posted by: John Baez on July 20, 2019 7:04 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

John wrote:

locally compact Hausdorff abelian topological groups. Of these, the nicest are the abelian Lie groups

To be honest, my usual stock of examples of locally compact abelian groups is no bigger than $\mathbb{R}^n$, $\mathbb{T}^n$, $\mathbb{Z}^n$, and finite abelian groups. (And, I suppose, products of these.) These are all Lie.

What other important examples of locally compact abelian groups are there, Lie or not?

The only new one that Wikipedia gives me is $\mathbb{Q}_p$.

Posted by: Tom Leinster on July 21, 2019 12:14 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Tom wrote:

What other important examples of locally compact abelian groups are there, Lie or not?

I feel like there’s a lot to say here, and I’m not sure where to start. But one starting point might be to “classify” all the Lie examples, and also consider their Pontryagin duals. Or maybe it’s better to start from the dual side?

Call a topological group $G$ compactly generated if there is a compact neighborhood of the identity that generates $G$ as a group. The compactly generated locally compact (Hausdorff) abelian groups are known: they are groups of the form

$K \times \mathbb{R}^m \times \mathbb{Z}^n$

where $K$ is a compact abelian group and $m, n$ are natural numbers.

The Pontryagin duals of compactly generated LCA groups are of the form

$A \times \mathbb{R}^m \times \mathbb{S}^n$

where $A$ is discrete and $S = S^1$. These are Lie LCA groups, and they are all the Lie LCA groups. So the compactly generated LCA groups and Lie LCA groups are dual to one another.

The intersection of these classes (i.e., compactly generated Lie LCA groups) are called elementary Lie groups. This is the class Tom mentioned: it consists of groups of the form

$F \times S^k \times \mathbb{R}^m \times \mathbb{Z}^n$

where $F$ is a finite abelian group.

Every LCA group is a filtered colimit of the system of its open compactly generated LCA subgroups and open inclusions between them. Dually, every LCA group is a cofiltered limit of Lie LCA groups.

Similarly, every Lie LCA group is a filtered colimit of elementary Lie groups and open inclusions between them, and every compactly generated LCA group is a cofiltered limit of elementary Lie groups.

Thus, all LCA groups are obtained by a process of starting with Tom’s class, and closing up under appropriate limits and colimits. For example, $\mathbb{Q}_p^\wedge$ is a filtered colimit of open inclusions

$\mathbb{Z}_p^\wedge \stackrel{p}{\to} \mathbb{Z}_p^\wedge \stackrel{p}{\to} \ldots$

where each map is multiplication by the prime $p$, and $\mathbb{Z}_p^\wedge$ is an inverse limit of finite discrete groups of type $\mathbb{Z}/(p^n)$ in the well-known way. The construction of the adeles by a similar procedure is only slightly more complicated.

I don’t know a lot about solenoids, but they are said to be the beginning of the collaboration between Eilenberg and Mac Lane. (I think Mac Lane was studying cohomology and group extensions and was to give a talk on this that Eilenberg said he had to miss, so he asked Mac Lane to give him the talk beforehand, and after hearing it said something to the effect that it smells like something we do in topology with $p$-adic solenoids, and the two stayed up all night to find a connection, and it all started there.) Solenoids do remind me a little of Bohr compactifications of LCA groups. Explicitly, take an LCA group $G$, pass to its Pontryagin dual, discretize that, and then take the Pontryagin dual of that: that’s $Bohr(G)$. All such Bohr compactifications are inverse limits of copies of $S^1$, although I guess usually the inverse system will be much larger than a simple inverse $\omega$-chain.

Returning a bit to analysis: not very long ago I wanted to explain to myself what Schwartz-Bruhat functions on LCA groups are. These are something like the rapidly vanishing smooth functions on elementary Lie groups, that one uses to get Schwartz space and its dual consisting of tempered distributions, except that this is now extended to all LCA groups. If you’re interested, you can see what I came up with here. I found it very difficult though to extract this description from the literature, because most of it from what I’ve seen doesn’t take good advantage of categorical concepts to give a clean account. However, some of the structure theory adumbrated above enters into the description.

Posted by: Todd Trimble on July 21, 2019 11:24 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

That’s very helpful; thanks. I’m also gratified to know that my usual stock of examples of LCAGs was not just some ragtag randomly-acquired collection but, by chance, also something important with a name: the elementary Lie groups.

Your account also helps me to place $\mathbb{Q}_p$ on the one hand, and solenoids on the other, in the overall landscape of LCAGs.

Posted by: Tom Leinster on July 22, 2019 2:28 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Solenoid groups. Some example discussions covering different aspects: Wikipedia, Terry Tao (see Exercise 6), Paul Garrett, Matt E on math.stackexchange (where he identifies a solenoid with the adele class group).

Posted by: David Roberts on July 21, 2019 12:54 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

And, so, I guess abelian varieties over the adeles should also give locally compact abelian topological groups…

Posted by: David Roberts on July 21, 2019 12:56 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

I wrote:

What other important examples of locally compact abelian groups are there, Lie or not?

Solenoid groups.

Thank you, David. Can you explain what makes solenoid groups important?

To be clear, I’m not asking for a complete list of locally compact abelian groups. I’m sure there are a ton other than the ones I mentioned. What I’m asking is for ones that are in some way important, and have a different feel from $\mathbb{R}^n$, $\mathbb{Z}^n$, $\mathbb{T}^n$ and finite groups. For instance, $\mathbb{Q}_p$ is one which I hadn’t consciously noted before. There must be others.

The stuff in the Wikipedia article about solenoids as dynamical attractors seems like it could be important, but I couldn’t see what it might say about solenoids as locally compact abelian groups — algebraic structures, as opposed to mere subsets of $\mathbb{R}^3$. Do you know?

Posted by: Tom Leinster on July 21, 2019 2:13 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

PS - I do appreciate that you gave copious links to different aspects of solenoids, so it’s a bit cheeky of me to ask for you to give an explanation too. Nevertheless, sometimes all one has space for is a small snack rather than a three-course meal.

It’s not the definition of solenoid that needs explaining. That’s simple enough: it’s just the limit in the category of locally compact abelian groups of a diagram of the form

$\cdots \to S^1 \to S^1 \to S^1,$

where the arrows are any homomorphisms you like. The only endomorphisms of $S^1$ are multiplication by an integer (or integer powers, if you’re writing the group operation as multiplication), so the possibilities are fairly limited. And typically all the homomorphisms in the diagram are taken to be the same.

But what I don’t get is why solenoids are important. If you have a short explanation that doesn’t assume anything about adeles etc., that would be great.

Posted by: Tom Leinster on July 21, 2019 2:22 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

I think they are a good example just beyond the familiar abelian Lie groups, and Terry Tao seems to use them as a source of reality checks in his notes on Hilbert’s fifth problem: given locally compact abelian but not locally Euclidean, what works and what breaks?

Other than that, as Todd pointed out, solenoids are historically important as being the catalyst for E and M L to get together and then invent category theory.

And, for the sake of lurkers, adeles (which I can’t claim to know about) are important for number theory/Langlands/arithmetic geometry, and probably are the type of thing that turns up in work of the Scholze school. So being locally compact abelian is about the nicest thing one can say about such groups, as otherwise they are kinda pathological from the point of view of people who are more used to differential geometry (but the p-adic people would disagree).

Posted by: David Roberts on July 22, 2019 12:53 AM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Mike and Tom wrote:

The short answer is that the Laplace transform is really just a generalization of the familiar Laurent series representation of complex analytic functions, but where the exponents are allowed to be non-integers and to “vary continuously” rather than discretely

This immediately made me think of the Fourier transform as a kind of generalization of Fourier series: you could replace both “Laplace” and “Laurent” by “Fourier” in your sentence above, and it would still more or less be true.

I’m not sure that it’s important at all here, but it’s interesting to note that Fourier series and Laurent series are really the same thing: just let the unit circle $\mathbb{T}$ be $\{ z \in \mathbb{C} : | z | = 1 \}$ instead of $\mathbb{R} / 2\pi \mathbb{Z}$ (as Tom implicitly used) and write $z^k$ instead of $e^{ 2 \pi i k}$. We just use different names in different contexts: “Laurent” when we’re interested in very nice functions that happen to be defined on a larger domain than just $\mathbb{T}$, and “Fourier” when we’re interested in less nice functions and don’t necessarily care about values on a larger domain.

So the Laplace and Fourier transforms generalize the same kind of gadget in slightly different directions. (Well, in orthogonal directions, actually!)

Posted by: Mark Meckes on July 22, 2019 4:30 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Of course this point is basically already in Mike’s post. That’s what I get for responding to a comment after only skimming the original post.

Posted by: Mark Meckes on July 22, 2019 4:32 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

I’ve mentioned before some work by Vitali Milman and collaborators (some more recent than the works I cited in that old comment) finding axiomatic characterizations of various important transformations in analysis. My favorite is a result he proved with Shiri Arstein-Avidan and Semyon Alesker characterizing the Fourier transform as essentially the unique transformation of one function into another that turns pointwise products into convolutions, and vice-versa. (I’ve left out some technical details there, but surprisingly few!)

As far as I know they haven’t proved any such result for the Laplace transform, but presumably such a result would give another satisfying answer to the question, “What is the Laplace transform?”

Posted by: Mark Meckes on July 22, 2019 4:28 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Seems at one point you are describing the inverse Mellin transform.

Posted by: Tom Copeland on September 15, 2019 3:47 PM | Permalink | Reply to this

### Re: What is the Laplace Transform?

Yes, I think the (inverse) Mellin transform is at least roughly another way of coordinatizing $H$.

Posted by: Mike Shulman on September 16, 2019 9:20 PM | Permalink | Reply to this

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