Spectral analysis and integral representations of the tempered fractional Riesz derivative

In this work, we develop the harmonic analysis associated with the second-order differential operator ${\mathcal{L}}_{\gamma }=-\frac{d^2}{d{x}^2}-2\gamma \frac{d}{dx}-{\gamma }^2$. Fractional powers of ${\mathcal{L}}_{\gamma }$ are defined via spectral representation, and a singular integral representation is provided. Furthermore, we establish the equivalence between the fractional powers of ${\mathcal{L}}_{\gamma }$ and a tempered Riesz derivative.