Quaternion and Clifford Fourier Transforms 1st Edition by Eckhard Hitzer 1000429369 9781000429367

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ISBN 10: 1000429369
ISBN 13: 9781000429367
Author: Eckhard Hitzer

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.). Features 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

Quaternion and Clifford Fourier Transforms 1st Table of contents:

Chapter 1: Introduction

• Historical Context and Definitions: Introduces the concepts and history behind quaternion Fourier transforms (QFT) and Clifford’s geometric algebra.
• Quaternion and Clifford Wavelets: Discusses wavelet theory within the context of quaternion and Clifford analysis.

Chapter 2: Clifford Algebra

• Axioms and Structure: Describes the foundational axioms of Clifford algebra and examples of its applications in geometry (e.g., rotations, reflections).
• Quadratic Forms: Provides an exploration of quadratic forms and their relationships to Clifford algebras.
• Geometric Products: Discusses the fundamental operations in Clifford algebra, such as the geometric, scalar, and outer products.
• Geometric Algebra Applications: Includes a discussion on the role of Clifford algebra in various geometric modeling and spacetime structures.

Chapter 3: Geometric Calculus

• Vector and Differential Calculus: Introduces vector calculus as applied to geometric algebra.
• Geometric Algebra for Differential Calculus: Focuses on the use of geometric algebra in differential calculus and its properties.

Chapter 4: Quaternion Fourier Transforms

• Fundamentals of QFT: Covers the basics of quaternion Fourier transforms and the ±-split, which is key in understanding transformations in quaternion space.
• Properties of QFT: Discusses uncertainty principles, convolutions, and special transforms like the windowed QFT and the quaternionic Fourier Mellin transform.
• Applications of QFT: Includes theoretical discussions as well as practical examples and properties.

Chapter 5: Clifford Fourier Transforms

• Overview of CFT: Introduces the Clifford Fourier transform and explores its properties, applications in differential calculus, and convolution.
• One-Sided and Two-Sided CFT: Explores both one-sided and two-sided Fourier transforms in Clifford algebras.
• Special Clifford Fourier Transforms: Includes specific cases like windowed Clifford Fourier transforms and Clifford Fourier Mellin transforms

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Eckhard Hitzer,Quaternion,Clifford Fourier