Fourier transform on the Lobachevsky plane and operational calculus

DOI10.1134/S001626632004005XarXiv2006.11581MaRDI QIDQ6343340

Yu. A. Neretin

Publication date: 20 June 2020

Abstract: The classical Fourier transform on the line sends the operator of multiplication by $x$ to $ifracddxi$ and the operator of differentiation $fracddx$ to the multiplication by $-ixi$. 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.

Mathematics Subject Classification ID

Harmonic analysis on homogeneous spaces (43A85) Analysis on real and complex Lie groups (22E30) Semisimple Lie groups and their representations (22E46)

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