ABSTRACT In 1990, van Eijndhoven and Meyers provide a special orthonormal basis for the Bargmann Hilbert space consisting of holomorphic Hermite functions. Then it was be natural to look for its orthogonal complement in the underlying -Hilbert space. In this paper, we describe the orthogonal complement of this Hilbert space. More precisely, a polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal–Bargmann integral transforms are provided. The obtained basis are then used to provide a non-trivial - and -fractional like-Fourier transforms.

中文翻译：

---

全纯 Hermite 函数和相关的 1d 和 2d 分数阶傅立叶变换的特殊正交补基

摘要 1990 年，van Eijndhoven 和 Meyers 为由全纯 Hermite 函数组成的 Bargmann Hilbert 空间提供了一个特殊的正交基。然后很自然地在底层 -Hilbert 空间中寻找它的正交补集。在本文中，我们描述了这个 Hilbert 空间的正交补。更准确地说，给出了多解析标准正交基，并提供了相应再现核函数和 Segal-Bargmann 积分变换的显式表达式。然后使用获得的基础来提供非平凡的和分数阶的傅立叶变换。