Fourier Transform on the Lobachevsky Plane and Operational Calculusстатья
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Аннотация:The classical Fourier transform on the line sends the operator of multiplication by $x$ to $\frac{id}{d\xi}$ and the operator $\frac{d}{dx}$ of differentiation to multiplication by $-i\xi$. For the Fourier transform on the Lobachevsky plane, we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier image, where shift operators act in the imaginary direction, i.e., a direction transversal to the integration contour in the Plancherel formula.