What is the meaning of this equation? Has it a name? $$f\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)y(x)=\mathcal{F}^{-1}_t\left[f(2\pi i t)\cdot\mathcal{F}_x[y(x)](t)\right](x)$$ Where $\mathcal{F}_x[y(x)](t)$ is the Fourier transform of $y(x)$
Special case: if $f(z)=\exp(az)$
$$\exp\left(a\frac{\mathrm{d}}{\mathrm{d}x}\right)y(x)=\mathcal{F}^{-1}_t\left[\exp(2\pi i a t)\cdot\mathcal{F}_x[y(x)](t)\right](x)\\
e^{a\frac{\mathrm{d}}{\mathrm{d}x}}y(x)=\mathcal{F}_t^{-1}\left[e^{2\pi ia t}\cdot\mathcal{F}_x[y(x)](t)\right](x)\\
e^{a\frac{\mathrm{d}}{\mathrm{d}x}}y(x)=\mathcal{F}_t^{-1}\left[\mathcal{F}_x[y(x)](t)\right](x+a)\\
e^{a\frac{\mathrm{d}}{\mathrm{d}x}}y(x)=y(x+a)
$$
Is the shift operator.