A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Ito--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Ito--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established

中文翻译：

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与 Itô-Hermite 多项式相关的二维分数阶傅里叶变换的对偶

一类积分变换在平面高斯希尔伯特空间上，其范围在双盘上的加权伯格曼空间中，被定义为与伊藤-埃尔米特多项式的梅勒函数相关的二维分数阶傅立叶变换的对偶变换。研究了这些变换的一些光谱特性。也就是说，我们研究它们的有界性并识别它们的零空间以及它们的范围。这种识别依赖于 Ito--Hermite 多项式的零集。此外，给出了它们奇异值的显式表达式，并研究了 p-Schatten 类的紧性和隶属度。还建立了与特定分数 Hankel 变换的关系