Can a gaussian be expressed as a sum of sines or cosines?

Question

I'm taking a quantum mechanics course in university right now and we're dealing with gaussian wave packets and I'm particularly interested in whether a gaussian can be expressed as a sum of sines or cosines (so that I could find the group and phase velocity of these wave packets)?

Answer 1

Taking for example the function

$$f(x)=e^{-x^2}\tag{1}$$

which I believe can be evaluated as

$$f(x)=\underset{N, f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{\mu(2 n-1)}{2 n-1} \left(\frac{3 F(0)}{8}\\+\sum _{k=1}^{2 f (2 n-1)} (-1)^k\, \cos\left(\frac{\pi k}{2 n-1}\right)\, F\left(\frac{k}{4 n-2}\right)\, \cos\left(\frac{\pi k x}{2 n-1}\right)\\-\frac{1}{4} \sum\limits_{k=1}^{4 f (2 n-1)} (-1)^k\, F\left(\frac{k}{8 n-4}\right)\, \cos\left(\frac{\pi k x}{4 n-2}\right)\right)\right)\tag{2}$$

where $\mu(n)$ is the Möbius function, the evaluation frequency $f$ in the two inner sums over $k$ is assumed to be a positive integer, and

$$F(\omega)=\mathcal{F}_x[f(x)](\omega)=\int\limits_{-\infty}^{\infty} e^{-x^2} e^{-2 \pi i \omega x} \, dx=\sqrt{\pi}\, e^{-\pi^2 \omega^2}\tag{3}$$

is the Fourier transform of $f(t)$.

---

My related MSO question provides information on the derivation of formula (2) above (which corresponds to formula (11) in my related MSO question).

---

Figures (1) to (3) below illustrate formula (2) for $f(x)=e^{-x^2}$ in orange overlaid on the blue reference function $f(x)$ where formula (2) is evaluated at $f=4$ and $N=10$. Since $f(x)$ is complex analytic I believe formula (2) actually converges globally for $x\in\mathbb{C}$.

---

Figure (1): Illustration of formula (2) for $f(x)$ in orange overlaid on the blue reference function $f(x)$

---

Figure (2): Illustration of real part of formula (2) for $f(1+i t)$ in orange overlaid on the blue reference function $\Re(f(1+i t))$

---

Figure (3): Illustration of imaginary part of formula (2) for $f(1+i t)$ in orange overlaid on the blue reference function $\Im(f(1+i t))$

---