In Quantum Mechanics, we consider firstly:
\[
\sum_{n=-\infty}^\infty a_n e^{2\pi i n x} = F(x)\\
\]
Where by, the value \(a_n\) is written as:
\[
a_n = \int_0^{1} F(x) e^{-2\pi i n x}\,dx\\
\]
We have a perfect interchange between \(a_n\) and \(F(x)\)... how I've just written it, is just Fourier series. Let's advance this to the fourier transform:
\[
\hat{F}(s) = \int_{-\infty}^\infty F(x) e^{-2\pi i x s}\,dx\\
\]
Where:
\[
F(x) = \int_{-\infty}^\infty \hat{F}(s) e^{2\pi i x s}\,ds\\
\]
This is just the basic precept of \(n \mapsto F(x)\) and \(F(x) \mapsto n\), but it does so in an unbelievably convenient manner. I could go on about Gottfried's work and how it relates to Heisenberg's work--or how my own work relates deeply to Von Neumann/Schrodinger/Feynman shit. But I won't.
Instead I will write that: The fourier transform acts on \(\mathbb{R}\) as an additive thing. When you evolve into algebraic fourier transforms, we get the mellin transform firstly. This is the "fourier transform on the semi-group \(\{\mathbb{R}^+, \times\}\)". Which means we can take \(x \mapsto x^m\), instead of \(x \mapsto mx\). The Mellin transform is a deeply connected object to fourier transforms, but it relates to "multiplication" more than fourier transforms. It acts upon the group \(\{\mathbb{R}^+, \times\}\), rather than the fourier transform acting upon \(\{\mathbb{R},+\}\)...
This gets very difficult though. And Ramanujan gave a very beautiful result, for the multiplicative case. This is known as Ramanujan's Master Theorem. It's in many ways a fancy reduction of Poisson's summation formula. But there's an exception to it. No such formula exists in Fourier analysis--similar results arise, but nothing exactly like that. But nonetheless, this is still Fourier analysis; just instead of additive fourier analysis, we are doing multiplicative.
It helps at this point to understand fourier transforms on algebraic groups. We can define a fourier transform on any group \(\mathbb{G} = \{G, \times\}\), so long as the group satisfies sufficient criteria. This is the basis of Tate's Thesis, and what won him the fields' medal in the fuckin 70s. The "usual fourier transform" is the fourier transform on the group \(\{\mathbb{R},+\}\). The fourier transform on the group \(\{\mathbb{R}^+,\times\}\) is the mellin transform...
We can write Fourier series as Fourier transforms, so this is a strict generalization.
The Mellin transform is a Fourier Transform........ It's just a few variable changes atop the fourier transform. So think of \(\vartheta\) as \(F\), and \(f^{\circ s}\) as \(\widehat{F}\). But \(\vartheta\) exists in the space \(\{\mathbb{R}^+,\times\}\), and so does \(f^{\circ s}\). This just requires a change of variables: \(F(e^x) = \vartheta(x)\) (sorry this is wrong)
Sorry went too fast, the change of variables is a little trickier:
\[
\begin{align}
\vartheta(e^x) &= F(x)\\
f^{\circ s} &= \int_{-\infty}^\infty F(x) e^{-xs}\,dx\\
s &\mapsto 2\pi i s\\
f^{\circ 2 \pi i s} &= \widehat{F}(s)\\
\end{align}
\]
But still, the quantum physics stuff obviously appears. But not really quantum physics, just the math from quantum physics...
EDIT: If you are interested in the Fourier transform, especially as I use it. I highly suggest Stein & Shakarchi's Complex Analysis; and specificly their chapter on the Fourier Transform. You can note that they describe \(F(x)\) holomorphic for \(a<|\Im(x)| < b\), as a primary identifier (where we have some kind of decay as \(|\Re x| \to \infty\)). This is no different then how I consider \(\vartheta(x)\) being differintegrable on a sector \(|\arg(-x)| < \kappa\). All we've done is a change of variables \(\vartheta(e^x) = F(x)\). In my case I work on special cases, where \(F(-\infty) = 0\) is satisfied on the Riemann sphere; which equates to \(\vartheta(x)\) is holomorphic at \(x=0\). That allows for Ramanujan's master theorem--which is really a super fancy Poisson Summation formula type thing, lol.
\[
\sum_{n=-\infty}^\infty a_n e^{2\pi i n x} = F(x)\\
\]
Where by, the value \(a_n\) is written as:
\[
a_n = \int_0^{1} F(x) e^{-2\pi i n x}\,dx\\
\]
We have a perfect interchange between \(a_n\) and \(F(x)\)... how I've just written it, is just Fourier series. Let's advance this to the fourier transform:
\[
\hat{F}(s) = \int_{-\infty}^\infty F(x) e^{-2\pi i x s}\,dx\\
\]
Where:
\[
F(x) = \int_{-\infty}^\infty \hat{F}(s) e^{2\pi i x s}\,ds\\
\]
This is just the basic precept of \(n \mapsto F(x)\) and \(F(x) \mapsto n\), but it does so in an unbelievably convenient manner. I could go on about Gottfried's work and how it relates to Heisenberg's work--or how my own work relates deeply to Von Neumann/Schrodinger/Feynman shit. But I won't.
Instead I will write that: The fourier transform acts on \(\mathbb{R}\) as an additive thing. When you evolve into algebraic fourier transforms, we get the mellin transform firstly. This is the "fourier transform on the semi-group \(\{\mathbb{R}^+, \times\}\)". Which means we can take \(x \mapsto x^m\), instead of \(x \mapsto mx\). The Mellin transform is a deeply connected object to fourier transforms, but it relates to "multiplication" more than fourier transforms. It acts upon the group \(\{\mathbb{R}^+, \times\}\), rather than the fourier transform acting upon \(\{\mathbb{R},+\}\)...
This gets very difficult though. And Ramanujan gave a very beautiful result, for the multiplicative case. This is known as Ramanujan's Master Theorem. It's in many ways a fancy reduction of Poisson's summation formula. But there's an exception to it. No such formula exists in Fourier analysis--similar results arise, but nothing exactly like that. But nonetheless, this is still Fourier analysis; just instead of additive fourier analysis, we are doing multiplicative.
It helps at this point to understand fourier transforms on algebraic groups. We can define a fourier transform on any group \(\mathbb{G} = \{G, \times\}\), so long as the group satisfies sufficient criteria. This is the basis of Tate's Thesis, and what won him the fields' medal in the fuckin 70s. The "usual fourier transform" is the fourier transform on the group \(\{\mathbb{R},+\}\). The fourier transform on the group \(\{\mathbb{R}^+,\times\}\) is the mellin transform...
We can write Fourier series as Fourier transforms, so this is a strict generalization.
The Mellin transform is a Fourier Transform........ It's just a few variable changes atop the fourier transform. So think of \(\vartheta\) as \(F\), and \(f^{\circ s}\) as \(\widehat{F}\). But \(\vartheta\) exists in the space \(\{\mathbb{R}^+,\times\}\), and so does \(f^{\circ s}\). This just requires a change of variables: \(F(e^x) = \vartheta(x)\) (sorry this is wrong)
Sorry went too fast, the change of variables is a little trickier:
\[
\begin{align}
\vartheta(e^x) &= F(x)\\
f^{\circ s} &= \int_{-\infty}^\infty F(x) e^{-xs}\,dx\\
s &\mapsto 2\pi i s\\
f^{\circ 2 \pi i s} &= \widehat{F}(s)\\
\end{align}
\]
But still, the quantum physics stuff obviously appears. But not really quantum physics, just the math from quantum physics...
EDIT: If you are interested in the Fourier transform, especially as I use it. I highly suggest Stein & Shakarchi's Complex Analysis; and specificly their chapter on the Fourier Transform. You can note that they describe \(F(x)\) holomorphic for \(a<|\Im(x)| < b\), as a primary identifier (where we have some kind of decay as \(|\Re x| \to \infty\)). This is no different then how I consider \(\vartheta(x)\) being differintegrable on a sector \(|\arg(-x)| < \kappa\). All we've done is a change of variables \(\vartheta(e^x) = F(x)\). In my case I work on special cases, where \(F(-\infty) = 0\) is satisfied on the Riemann sphere; which equates to \(\vartheta(x)\) is holomorphic at \(x=0\). That allows for Ramanujan's master theorem--which is really a super fancy Poisson Summation formula type thing, lol.