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

IF 1.1 2区 数学 Q2 MATHEMATICS, APPLIED
Lakshmanan Subbiah, Roopkumar Rajakumar
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引用次数: 0

Abstract

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.

多维广义分式 $${p\mb {S}}$ 变换
本文介绍了一种新的多维分数 S 变换(S_{\phi ,\varvec{\alpha},\lambda }\),它使用了广义分数卷积(\star _{\varvec\{alpha },\lambda }\)和满足某些可接受性条件的广义窗函数(\phi \)。\(S_{\phi ,\varvec{\alpha },\lambda }f\) 的值也可以写成输入函数 f 与合适函数 \(\phi _{textbf{t},\textbf{u}}^{\varvec\{alpha }_{\lambda }}\) 的内积形式。用广义分数卷积来表示 \(S_{\phi ,\varvec{\alpha },\lambda }f\) 可以帮助我们利用广义分数卷积定理得到 \(S_{\phi ,\varvec{\alpha },\lambda }\) 的帕瑟瓦尔公式。然后,根据帕瑟瓦尔特性证明了反转定理。在 \(S_{\phi ,\varvec\{alpha },\lambda }\) 的核中使用广义窗函数,可以选择窗函数的傅里叶变换为紧凑支撑的平滑函数或快速递减函数。我们还讨论了 \(L^2(\mathbb {R}^N, \mathbb {C})\) 上 \(S_\{phi ,\varvec{\alpha },\lambda }\) 范围的特征。最后,我们将变换扩展到一类四元数值函数。
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来源期刊
Advances in Applied Clifford Algebras
Advances in Applied Clifford Algebras 数学-物理:数学物理
CiteScore
2.20
自引率
13.30%
发文量
56
审稿时长
3 months
期刊介绍: Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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