Geodesic
Fourier Transform on the Lobachevsky Plane and Operational Calculus
Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 64-73

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The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator $\frac{d}{d x}$ 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.
Keywords: group $\operatorname{SL}(2,\mathbb{R})$, representations of the principal series, differential-difference operators.
Mots-clés : Plancherel decomposition 
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     author = {Yu. A. Neretin},
     title = {Fourier {Transform} on the {Lobachevsky} {Plane} and {Operational} {Calculus}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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Yu. A. Neretin. Fourier Transform on the Lobachevsky Plane and Operational Calculus. Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 64-73. http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a4/