Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is defined by the function
$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$
and $f(y)$ can be recovered by the Mellin inversion formula:
$$f(y) = \frac{1}{2\pi i} \int_{\sigma – i \infty}^{\sigma + i \infty} y^{-s} M(f)(s) ds.$$
This is a change of variable from the Fourier inversion formula, or the Laplace inversion formula, and can be proved in the same way. This is used all the time in analytic number theory (as well as many other subjects, I understand) — for example, if $f(y)$ is the characteristic function of $[0, 1]$ then its Mellin transform is $1/s$, and one recovers the fact (Perron's formula) that
$$\frac{1}{2\pi i} \int_{2 – i \infty}^{2 + i \infty} n^{-s} \frac{ds}{s}$$
is equal to 1 if $0 < n < 1$, and is 0 if $n > 1$. (Note that there are technical issues which I am glossing over; one integrates over any vertical line with $\sigma > 0$, and the integral is equal to $1/2$ if $n = 1$.)
I use these formulas frequently, but… I find myself having to look them up repeatedly, and I'd like to understand them more intuitively. Perron's formula can be proved using Cauchy's residue formula (shift the contour to $- \infty$ or $+ \infty$ depending on whether $n > 1$), but this proof doesn't prove the general Mellin inversion formula.
My question is:
What do the Mellin transform and the inversion formula mean? Morally, why are they true?
For example, why is the Mellin transform an integral over the positive reals, while the inverse transform is an integral over the complex plane?
I found some resources — Wikipedia; this MO question is closely related, and the first video in particular is nice; and a proof is outlined in Iwaniec and Kowalski — but I feel that there should be a more intuitive explanation than any I have come up with so far.
Best Answer
[Some next-day edits in response to comments] As counterpoint to other viewpoints, one can say that Mellin inversion is "simply" Fourier inversion in other coordinates. Depending on one's temperament, this "other coordinates" thing ranges from irrelevancy to substance... The question about moral imperatives for Fourier inversion is addressed a bit below.
[Added: the exponential map $x\rightarrow e^x$ gives an isomorphism of the topological group of additive reals to multiplicative. Thus, the harmonic analysis on the two is necessarily "the same", even if the formulaic aspects look different. The occasional treatment of values (and derivatives) at $0$ for functions on the positive reals, as in "Laplace transforms", is a relative detail, which certainly has a corresponding discussion for Fourier transforms.]
The specific riff in Perron's identity in analytic number theory amounts to (if one tolerates a change-of-coordinates) guessing/discerning an L^1 function on the line whose Fourier transform is (in what function space?!) the characteristic function of a half-line.
Since the char fcn of a half-line is not in L^2, and does not go to 0 at infinity, there are bound to be analytical issues... but these are technical, not conceptual.
[Added: the Fourier transform families $x^{\alpha-1}e^{-x}\cdot \chi_{x>0}$ and $(1+ix)^{-\alpha}$, (up to constants) where $\chi$ is the characteristic function, when translated to multiplicative coordinates, give one family approaching the desired "cut-off" effect of the Perron integral. There are other useful families, as well.]
To my taste, the delicacies/failures/technicalities of not-quite-easily-legal aspects of Fourier transforms are mostly crushed by simple ideas about Sobolev spaces and Schwartz' distributions... tho' these do not change the underlying realities. They only relieve us of some of the burden of misguided fussiness of some self-appointed guardians of a misunderstanding of the Cauchy-Weierstrass tradition.
[Added: surely such remarks will strike some readers as inappropriate poesy... but it is easy to be more blunt, if desired. Namely, in various common contexts there is a pointless, disproportionate emphasis on "rigor". Often, elementary analysis is the whipping-boy for this impulse, but also one can see elementary number theory made senselessly difficult in a similar fashion. Supposedly, the audience is being made aware of a "need/imperative" for care about delicate details. However, in practice, one can find oneself in the role of the Dilbertian "Mordac the Preventer (of information services)" [see wiki] proving things like the intermediate value theorem to calculus students: it is obviously true, first, or else one's meaning of "continuous" or "real numbers" needs adjustment; nevertheless, the traditional story is that this intuition must be delegitimized, and then a highly stylized substitute put in its place. What was the gain? Yes, something foundational, but time has passed, and we have only barely recovered, at some expense, what was obviously true at the outset.
On another hand, Bochner's irritation with "distributions theory" was that it was already clear to him that things worked like this, and he could already answer all the questions about generalized functions... so why be impressed with Schwartz' "mechanizing" it? For me, the answer is that Schwartz arranged a situation so that "any idiot" could use generalized functions, whereas previously it was an "art". Yes, sorta took the fun out of it... but maybe practical needs over-rule preservation of secret-society clubbiness?]
Why should there be Fourier inversion? (for example...) Well, we can say we want such a thing, because it diagonalizes the operator $d/dx$ on the line (and more complicated things can be said in more complicated situations).
Among other things, this renders "engineering math" possible... That is, one can understand and justify the almost-too-good-to-be-true ideas that seem "necessary" in applied situations... where I can't help but add "like modern number theory". :)
[Added: being somewhat an auto-didact, I was not aware until relatively late that "proof" was absolutely sacrosanct. To the point of fetishism? In fact, we do seem to collectively value insightful conjecture and not-quite-justifiable heuristics, and interesting unresolved ideas offer more chances for engagement than do settled, ironclad, finished discussions. For that matter, the moments that one intuits "the truth", and then begins looking for reasons, are arguably more memorable, more fun, than the moments at which one has dotted i's and crossed t's in the proof of a not-particularly-interesting lemma whose truth was fairly obvious all along. More ominous is the point that sometimes we can see that something is true and works despite being unable to "justify" it. Heaviside's work is an instance. Transatlantic telegraph worked fine despite...]
In other words: spectral decomposition and synthesis. Who couldn't love it?!
[Added: and what recourse do we have than to hope that reasonable operators are diagonalizable, etc? Serre and Grothendieck (and Weil) knew for years that the Lefschetz fixed-point theorem should have an incarnation that would express zeta functions of varieties in terms of cohomology, before being able to make sense of this. Ngo (Loeser, Clucker, et alteri)'s proof of the fundamental lemma in the number field case via model theoretic transfer from the function field case is not something I'd want to have to "justify" to negativists!]