In this paper, we introduce a new multidimensional fractional S transform \(S_{\phi ,\varvec{\alpha },\lambda }\) using a generalized fractional convolution \(\star _{\varvec{\alpha },\lambda }\) and a general window function \(\phi \) satisfying some admissibility condition. The value of \(S_{\phi ,\varvec{\alpha },\lambda }f\) is also written in the form of inner product of the input function f with a suitable function \(\phi _{\textbf{t},\textbf{u}}^{\varvec{\alpha }_{\lambda }}\). The representation of \(S_{\phi ,\varvec{\alpha },\lambda }f\) in terms of the generalized fractional convolution helps us to obtain the Parseval’s formula for \(S_{\phi ,\varvec{\alpha },\lambda }\) using the generalized fractional convolution theorem. Then, the inversion theorem is proved as a consequence of the Parseval’s identity. Using a generalized window function in the kernel of \(S_{\phi ,\varvec{\alpha },\lambda }\) gives option to choose window function whose Fourier transform as a compactly supported smooth function or a rapidly decreasing function. We also discuss about the characterization of range of \(S_{\phi ,\varvec{\alpha },\lambda }\) on \(L^2(\mathbb {R}^N, \mathbb {C})\). Finally, we extend the transform to a class of quaternion valued functions consistently.

在本文中，我们引入了一种新的多维分数S变换\(S_{\phi ,\varvec{\alpha },\lambda }\)，使用广义分数卷积\(\star _{\varvec{\alpha },\ lambda }\)和满足某些可接受性条件的一般窗口函数\(\phi \) 。 \(S_{\phi ,\varvec{\alpha },\lambda }f\)的值也可以写成输入函数f与合适函数\(\phi _{\textbf{ t},\textbf{u}}^{\varvec{\alpha }_{\lambda }}\)。用广义分数卷积表示\ (S_{\phi ,\varvec{\alpha },\lambda }f\)有助于我们获得\(S_{\phi ,\varvec{\alpha },\lambda }\)使用广义分数卷积定理。然后，根据Parseval恒等式证明了反演定理。在\(S_{\phi ,\varvec{\alpha },\lambda }\)内核中使用广义窗函数可以选择窗函数，其傅立叶变换为紧支持的平滑函数或快速递减函数。我们还讨论了\(S_{\phi ,\varvec{\alpha },\lambda }\)在\(L^2(\mathbb {R}^N, \mathbb {C})\上的范围表征）。最后，我们一致地将变换扩展到一类四元数值函数。