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.

中文翻译：

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罗巴切夫斯基平面上的傅里叶变换和运算微积分

摘要

经典傅立叶就行变换由发送乘法的操作者\（X \）到\（I \压裂{d} {d \ XI} \）和操作者的\（\压裂{d} {DX} \）的乘以\(-i\xi\) 的微分。对于 Lobachevsky 平面上的傅里叶变换，我们建立了对某类微分算子的类似对应关系。看起来 Lobachevsky 平面上的微分算子对应于傅立叶图像中的微分-微分算子，其中移位算子作用于虚方向，即与 Plancherel 公式中的积分轮廓相交的方向。