Fourier Transform on the Lobachevsky Plane and Operational Calculus

Yu. A. Neretin

Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 64-73.

Résumé

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.

Détail

Keywords: group $\operatorname{SL}(2,\mathbb{R})$, representations of the principal series, differential-difference operators.
Mots-clés : Plancherel decomposition

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