Abstract
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.
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Notes
At a first glance, this condition seems awkward. In fact, it is more natural to consider the space of smooth functions (or spaces of smooth sections of linear bundles) on the circle (the projective line) \( \mathbb{R} \cup \infty\). Passing to a space of functions on the line, we cut the circle; for this reason, we must impose sewing conditions at infinity.
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This work was supported by the Austrian Science Fund (FWF), project no. P31591.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2020, Vol. 54, pp. 64-73 https://doi.org/10.4213/faa3812.
Translated by Yu. A. Neretin
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Neretin, Y.A. Fourier Transform on the Lobachevsky Plane and Operational Calculus. Funct Anal Its Appl 54, 278–286 (2020). https://doi.org/10.1134/S001626632004005X
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DOI: https://doi.org/10.1134/S001626632004005X