论文标题
Lobachevsky平面和操作演算上的傅立叶变换
Fourier transform on the Lobachevsky plane and operational calculus
论文作者
论文摘要
该行上的经典傅立叶变换将乘以$ x $的运算符发送到$ i \ frac {d} {dξ} $，而差异化$ \ frac {d} {d x} $的操作员则通过$ -i-iξ$。对于Lobachevsky平面上的傅立叶变换，我们为某个差异操作员家族建立了类似的对应关系。似乎Lobachevsky平面上的差分运算符对应于傅立叶图像中的差分差异操作员，其中移动操作员在假想方向上作用，即在PlanCherel公式中横向到集成轮廓的方向。
The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{dξ}$ and the operator of differentiation $\frac{d}{d x}$ 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.
