Thursday 27 March 2025
The Paley-Wiener theorem is a fundamental concept in mathematical analysis, and its applications are far-reaching. Recently, researchers have extended this theorem to the realm of slice regular functions, opening up new possibilities for signal processing and analysis.
For those unfamiliar, the Paley-Wiener theorem is a result that connects the Fourier transform of a function with its decay properties. In essence, it states that if a function decays rapidly enough at infinity, then its Fourier transform must be supported on a certain interval. This has significant implications for signal processing, as it allows us to analyze and manipulate signals in the frequency domain.
In the context of slice regular functions, this theorem takes on new significance. Slice regular functions are a type of analytic function that is defined over the quaternions, rather than the usual complex numbers. Quaternions are a four-dimensional algebraic structure that can be used to represent 3D rotations and other geometric transformations.
The extension of the Paley-Wiener theorem to slice regular functions has important implications for signal processing in 3D space. By analyzing signals in the quaternion frequency domain, researchers can gain valuable insights into their spatial properties and behavior.
One potential application is in the field of computer vision, where understanding the spatial structure of images is crucial for tasks such as object recognition and tracking. By applying the Paley-Wiener theorem to slice regular functions, researchers may be able to develop new algorithms that are more robust and efficient than existing methods.
Another potential application is in robotics and 3D modeling, where understanding the spatial properties of objects is essential for tasks such as motion planning and collision detection. The extended Paley-Wiener theorem could provide a powerful tool for analyzing and manipulating 3D objects in a more intuitive and flexible way.
The authors of this research have developed a new sampling theorem that is based on the extended Paley-Wiener theorem. This theorem provides a precise characterization of the Fourier transform of slice regular functions, and it has important implications for signal processing and analysis.
In particular, the new sampling theorem shows that slice regular functions can be reconstructed from their samples in a more efficient way than previously thought. This has significant implications for applications such as data compression and transmission, where reducing the amount of data required to represent a signal is crucial.
The authors have also developed a new framework for analyzing and processing slice regular signals, which is based on the extended Paley-Wiener theorem.
Cite this article: “Extending the Paley-Wiener Theorem to Slice Regular Functions: New Possibilities for 3D Signal Processing and Analysis”, The Science Archive, 2025.
Signal Processing, Fourier Transform, Slice Regular Functions, Quaternions, Computer Vision, Robotics, 3D Modeling, Data Compression, Transmission, Sampling Theorem
Reference: Yanshuai Hao, Pei Dang, Weixiong Mai, “Paley-Wiener Theorems For Slice Regular Functions” (2025).
