Fourier transform on the Lobachevsky plane and operational calculus

TL;DR

This paper extends the classical Fourier transform to the Lobachevsky plane, establishing a correspondence between differential operators and differential-difference operators involving shifts in the imaginary direction.

Contribution

It introduces a Fourier transform on the Lobachevsky plane that relates differential operators to differential-difference operators with imaginary shifts.

Findings

Differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier image.

Shift operators act in the imaginary direction, transversal to the integration contour.

The established correspondence generalizes classical Fourier transform properties to hyperbolic geometry.

Abstract

The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator of differentiation $\frac{d}{d x}$ to the 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.