Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with the Lebesgue measure $dx$, $\mathbb{Z}/2\mathbb{Z}$ with the counting measure, and $\mathbb{R}^\times$ with the measure $\frac{dx}{|x|}$ (i.e. each of the above three groups is endowed with a Haar measure).

Any of the three groups above is isomorphic to its own dual group, so if $G$ is any of those groups, then the Fourier transform on this group can be seen as a (up to a multiplicative constant) unitary operator on $L^2(G)$.

The Fourier transform can also be given by the following explicit formulas in each of the three cases:

Fourier transform $\mathcal{F}_{(\mathbb{R},+)}$ on $(\mathbb{R},+)$:

The Fourier transform $\mathcal{F}_{(\mathbb{R},+)}$ on $(\mathbb{R},+)$ is given by \begin{align*} \mathcal{F}_{(\mathbb{R},+)}f(x) = \int_{\mathbb{R}} f(y)e^{-ixy} \, dy \tag{FT1} \end{align*} for every $f \in L^2(\mathbb{R},dx) \cap L^1(\mathbb{R},dx)$ (here, I chose the normalisation constant of the transform such that $\frac{1}{\sqrt{2\pi}} \mathcal{F}_{(\mathbb{R},+)}$ is a unitary).

Fourier transform on $(\mathbb{Z}/2\mathbb{Z},+)$:

The Fourier transform $\mathcal{F}_{(\mathbb{Z}/2\mathbb{Z},+)}$ on $(\mathbb{Z}/2\mathbb{Z},+)$ is given by \begin{align*} \mathcal{F}_{(\mathbb{Z}/2\mathbb{Z},+)}f = \begin{pmatrix} e^{i\pi\cdot 0 \cdot 0} & e^{i\pi\cdot 0 \cdot 1} \\ e^{i\pi\cdot 1 \cdot 0} & e^{i\pi \cdot 1 \cdot 1} \end{pmatrix} f = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} f \tag{FT2} \end{align*} for each $f \in L^2(\mathbb{Z}/2\mathbb{Z}) = \mathbb{C}^2$ (where I chose the normalisation constant such that $\frac{1}{\sqrt{2}}\mathcal{F}_{(\mathbb{Z}/2\mathbb{Z},+)}$ is a unitary).

Fourier transform on $(\mathbb{R}^\times,\cdot)$:

Define a function $\delta: \mathbb{R}^\times \to \{0,1\}$ by $\delta(x) = 0$ for $x > 0$ and $\delta(x) = 1$ for $x < 0$. Then the Fourier transform $\mathcal{F}_{(\mathbb{R}^\times,\cdot)}$ on the group $(\mathbb{R}^\times, \cdot)$ is given by \begin{align*} \mathcal{F}_{(\mathbb{R}^\times, \cdot)} f(x) = \int_{\mathbb{R}^\times} f(y) e^{-i\ln|x|\ln|y|} e^{i\pi \delta(x)\delta(y)} \frac{dy}{|y|} \tag{FT3} \end{align*} for all $f \in L^2(\mathbb{R}^\times, \frac{dx}{|x|}) \cap L^1(\mathbb{R}^\times,\frac{dx}{|x|})$ (here the normalisation is such that $\frac{1}{\sqrt{4\pi}} \mathcal{F}_{(\mathbb{R}^\times, \cdot)}$ is a unitary).

The formulas (FT1) and (FT2) are well-known and can be found in numerous textbooks and monographs. The formula (FT3) can be derived from (FT1) and (FT2) if one uses that the group $(\mathbb{R}^\times,\cdot)$ is isomorphic to the direct product of $(\mathbb{R},+)$ and $(\mathbb{Z}/2\mathbb{Z},+)$ via the mapping \begin{align*} \varphi: \mathbb{R}^\times \to \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}, \quad \varphi(x) = (\ln |x|, \delta(x) + 2 \mathbb{Z}). \end{align*}

Together with two colleagues I wish to use formula (FT3) in a research article directed towards an audience with a background mainly in stochastics and statistics, so we have to choose one of the following three options:

1) State formula (FT3) together with a reference to the literature.

2) State formula (FT3) together with a more or less explicit computation (possibly in an appendix) which explains/derives the formula.

3) State formula (FT3) and simply mention that its a consequence of the well-known formulas (FT1) and (FT3).

[In general, it might not be a bad idea either to combine 1) with 2) or 3).]

We dismissed option 2) because we do not want to include a lenghty computation about a well-known fact and because one has to use several isomorphisms in the computation if one wants to be precise -- which would force us to overload the computation with a lot of notation where actually nothing at all is happening. Option 3) would probably be a good choice for an audience with a strong background in harmonic analysis or operator theory, but it does not seem appropriate for an audience whose background is mainly in statistics. So we are left with 1), and we face the problem that I spend quite some time in the library browsing more than a dozen books about harmonic analysis, the Fourier transform and integral transforms without finding any reference which comes close to (FT3). So my question is:

Question: (a) Is there an explicit reference in the literature for (FT3)?

(b) If not, is there an explicit reference in the literature for a formula for the Fourier transform on the group $((0,\infty),\cdot)$ (endowed with the Haar measure $\frac{dx}{x}$)?

I would prefer mathematical literature to physical or engineering literature, but even a reference to a physics or engineering book would be much better than what we have at the moment (which is nothing).

Remark: There is an integral transform called the Mellin transform which can be seen as some kind of mixture between the Fourier transform on $(\mathbb{R},+)$ and $((0,\infty),\cdot)$. If $\mathcal{M}$ denotes the Mellin transform, then the mapping $f \mapsto (\mathcal{M}f)(-i\ln(\,\cdot\,))$ is the Fourier transform on $((0,\infty),\cdot)$. A reference to this fact would answer question (b) above, but again, I could not find such a reference in the literature.

To make this post more helpful to future visitors, the following extendend question seems worthwhile to be asked, too:

Extended question: Is there any reference in the (mathematical) literature where the Fourier transforms on most of the very classical groups such as $(\mathbb{R}^n,+)$, $(\mathbb{Z}^n,+)$, $(\mathbb{Z}/n\mathbb{Z},+)$, $(\mathbb{R}_{>0},\cdot)$, $(\mathbb{R}^\times, \cdot)$, $(\mathbb{T},\cdot)$ (the torus), etc., are explicitly listed in a somewhat encyclopedic way?

Disclaimer: Clearly, the formula (FT3) itself is not about research level mathematics. Yet, I'm hoping that the fact that I'm looking for an appropriate reference for (FT3) for a research article makes this reference request suitable for this site.

  • $\begingroup$ I would say that Mellin transform is what you are looking for: $f(x) \mapsto \mathcal{M} f(i \ln(x))$ is precisely the Fourier transform on $(\mathbb{R}^+,\cdot)$. $\endgroup$ Mar 19, 2018 at 18:51
  • $\begingroup$ @MateuszKwaśnicki: You are of course right that $f \mapsto \mathcal{M}f(i\ln(\,\cdot\,))$ is the Fourier transform on $(\mathbb{R}^+,\cdot)$ (that's why I mentioned the Mellin transform in the post). The problem is that we would like to give a reference in the article which says that exactly this is true (still, we would have to explain in this case how to switch from $(\mathbb{R}^+,\cdot)$ to $(\mathbb{R}^\times, \cdot)$). Of course, all those observations are quite straightforward if you have some experience with harmonic analysis (to be continued...) $\endgroup$ Mar 19, 2018 at 19:15
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    $\begingroup$ @MateuszKwaśnicki: But we suspect that our readership does not have this kind of experience, which is why we would prefer an explicit reference. $\endgroup$ Mar 19, 2018 at 19:16
  • $\begingroup$ @MateuszKwaśnicki: But admittedly, the wording "this is not quite what we are looking for" in my post somehow missed the point. I've edited the post and rephrased this. $\endgroup$ Mar 19, 2018 at 19:27
  • $\begingroup$ Sure, I understand what you mean. A quick search for "two-sided Mellin" only pointed me to: arXiv:1005.4712, a book and a video, none of which seems to satisfy your needs. $\endgroup$ Mar 19, 2018 at 20:58

1 Answer 1


For (a), the closest I’ve seen is Igusa (1974, pp. 112-113, 119) or Wolf (1979, §8.2).

For (b) and the extended question, you might prefer Folland (1992, §7.6).

  • $\begingroup$ Thank you for your answer! Similarly as the literature that I found the references you suggest are somewhat "implicit". For instance, Wolf discusses the Mellin transform but does not mention explicitly how it is related to the Fourier transform on $((0,\infty),\cdot)$. Folland discusses the Mellin transform and explains how it is related to the Fourier transform on $(\mathbb{R},+)$, but he does not explain how it is related to the Fourier transform on $((0,\infty),\cdot)$ (yet, as you wrote, Folland certainly gives a good overview of the Fourier transform on several classical groups). $\endgroup$ Mar 20, 2018 at 13:33
  • $\begingroup$ @JochenGlueck Well, they do something explicit. Any compelling reason you cannot use that over, say, (FT3)? That would save the (trivial) “transport of structure” argument you seem to want to avoid. $\endgroup$ Mar 20, 2018 at 21:39
  • $\begingroup$ You're right, those references are quite explicit in some respects (for instance, a version of Plancherel's theorem for the Mellin transform is stated). Maybe that's sufficient for our paper if we slightly change the structure of our argument. I'll discuss this with my co-authors. $\endgroup$ Mar 21, 2018 at 0:50

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