In this work, a pair of quadratic-phase Hankel (QPH) transforms is introduced and defined their inversion. Moreover, some differential operators are given, and two variants of Bessel differential operators are defined. Further two different types of Zemanian spaces are defined and discussed the continuity of QPH-transform and given differential operators on these spaces. Next, we extend the QPHT to generalized functions. QPH-translation and convolution are also defined and studied their properties. Furthermore, an application of QPH-transform to a generalized non-linear parabolic equation is given.

中文翻译：

---

一对二次相位 Hankel 变换的卷积

在这项工作中，引入了一对二次相位 Hankel (QPH) 变换并定义了它们的反演。此外，给出了一些微分算子，并定义了贝塞尔微分算子的两种变体。进一步定义了两种不同类型的 Zemanian 空间，并讨论了 QPH 变换的连续性，并在这些空间上给定了微分算子。接下来，我们将 QPHT 扩展到广义函数。还定义了 QPH 翻译和卷积并研究了它们的特性。此外，还给出了 QPH 变换在广义非线性抛物线方程中的应用。