Fourier transform on the Lobachevsky plane and operational calculus

Abstract

The classical Fourier transform on the line sends the operator of multiplication by x to i d dξ and the operator of differentiation d dx to the multiplication by −iξ. 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.