Are there examples of $f(x)$ for which $\mathcal{F}_x[f(x)](s)=\mathcal{F}_x^{-1}[f(x)](s)$ other than $f(x)=\cos(x)$ and $f(x)=\text{sinc}(x)$?

Question

I've noticed the Fourier and inverse Fourier transforms of $f(x)=\text{sinc}(x)$ (as well as $f(x)=\cos(x)$) are identical:

$$\mathcal{F}_x[\text{sinc}(x)](s)=\mathcal{F}_x^{-1}[\text{sinc}(x)](s)=\frac{1}{2}\,\sqrt{\frac{\pi }{2}} (\text{sgn}(1-s)+\text{sgn}(s+1))\tag{1}$$

$$\mathcal{F}_x[\cos(x)](s)=\mathcal{F}_x^{-1}[\cos(x)](s)=\sqrt{\frac{\pi}{2}} \,\delta(s-1)+\sqrt{\frac{\pi}{2}}\,\delta(s+1)\tag{2}$$

Question: Are there other examples of $f(x)$ for which $\mathcal{F}_x[f(x)](s)=\mathcal{F}_x^{-1}[f(x)](s)$, and is there a name for this class of functions?

Answer 1

I am going to assume that you are using the following definition of Fourier transform:

$$F(s)=\mathcal{F}_x[f(x)](s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-i x s} \;\mathrm{d}x.$$

The corresponding inverse Fourier transform is $$G(s)=\mathcal{F}_x^{-1}[f(x)](s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{i x s} \;\mathrm{d}x.$$

Note that $F(s)=G(-s)$, i.e., the inverse Fourier transform is just a time-reversed version of the direct Fourier transform.

Therefore, all functions that have an even Fourier transform (i.e., all even functions) have equal direct and inverse Fourier transforms with this definition.