Quaternion and Clifford Fourier Transforms describes the development of quaternion and Clifford Fourier transforms in Clifford (geometric) algebra over the last 30 years. It is the first comprehensive, self-contained book covering this vibrant new area of pure and applied mathematics in depth.

The book begins with a historic overview, followed by chapters on Clifford and quaternion algebra and geometric (vector) differential calculus (part of Clifford analysis). The core of the book consists of one chapter on quaternion Fourier transforms and one on Clifford Fourier transforms. These core chapters and their sections on more special topics are reasonably self-contained, so that readers already somewhat familiar with quaternions and Clifford algebra will hopefully be able to begin reading directly in the chapter and section of their particular interest, without frequently needing to skip back and forth. The topics covered are of fundamental interest to pure and applied mathematicians, physicists, and engineers (signal and color image processing, electrical engineering, computer science, computer graphics, artificial intelligence, geographic information science, aero-space engineering, navigation, etc.).


  • Intuitive real geometric approach to higher-dimensional Fourier transformations
  • A comprehensive reference, suitable for graduate students and researchers
  • Includes detailed definitions, properties, and many full step-by-step proofs
  • Many figures and tables, a comprehensive biography, and a detailed index make it easy to locate information

chapter Chapter 1|14 pages


chapter Chapter 2|90 pages

Clifford algebra

chapter Chapter 3|36 pages

Geometric calculus

chapter Chapter 4|100 pages

Quaternion Fourier transforms

chapter Chapter 5|146 pages

Clifford Fourier transforms

chapter Chapter 6|8 pages

On the interrelationship of QFTs and CFTs