Multidimensional Generalized Fractional \({\pmb {S}}\) Transform

In this paper, we introduce a new multidimensional fractional S transform \(S_{\phi ,\varvec{\alpha },\lambda }\) $S_{\varphi ,\mathit{\alpha },\lambda }$ using a generalized fractional convolution \(\star _{\varvec{\alpha },\lambda }\) ${\star }_{\mathit{\alpha },\lambda }$ and a general window function \(\phi \) $\varphi$ satisfying some admissibility condition. The value of \(S_{\phi ,\varvec{\alpha },\lambda }f\) $S_{\varphi ,\mathit{\alpha },\lambda }f$ is also written in the form of inner product of the input function f with a suitable function \(\phi _{\textbf{t},\textbf{u}}^{\varvec{\alpha }_{\lambda }}\) ${\varphi }_{\mathbf{t},\mathbf{u}}^{{\mathit{\alpha }}_{\lambda }}$ . The representation of \(S_{\phi ,\varvec{\alpha },\lambda }f\) $S_{\varphi ,\mathit{\alpha },\lambda }f$ in terms of the generalized fractional convolution helps us to obtain the Parseval’s formula for \(S_{\phi ,\varvec{\alpha },\lambda }\) $S_{\varphi ,\mathit{\alpha },\lambda }$ 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 \(S_{\phi ,\varvec{\alpha },\lambda }\) $S_{\varphi ,\mathit{\alpha },\lambda }$ 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 \(S_{\phi ,\varvec{\alpha },\lambda }\) $S_{\varphi ,\mathit{\alpha },\lambda }$ on \(L^2(\mathbb {R}^N, \mathbb {C})\) $L^2({\mathbb{R}}^N,\mathbb{C})$ . Finally, we extend the transform to a class of quaternion valued functions consistently.