In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant in the inequality for the quaternion offset linear canonical transform (QOLCT). We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner–Ville distribution associated with QQPFT and obtain the Lieb’s uncertainty and entropy uncertainty principles for these three transforms.

在本文中，我们将复值函数的二次相位傅里叶变换扩展到两个变量的四元数值函数。我们称之为四元数二次相位傅里叶变换（QQPFT）。基于QQPFT和四元数傅里叶变换（QFT）之间的关系，我们得到了QQPFT的尖锐Hausdorff-Young不等式，特别是尖锐了四元数偏移线性正则变换（QOLCT）不等式中的常数。我们定义了短时四元数二次相位傅里叶变换（STQQPFT）并探讨了它的一些性质，包括内积关系和反演公式。我们找到了它与二维四元数模糊函数和与 QQPFT 相关的四元数 Wigner-Ville 分布的关系，并获得了这三种变换的 Lieb 不确定性和熵不确定性原理。