Quaternion and Clifford Fourier Transforms

1. Introduction. 1.1. Brief Historical Notes. 1.2. Quaternion Fourier Transforms (QFT). 1.3. Clifford Fourier Transforms in Clifford's Geometric Algebra. 1.4. Quaternion and Clifford Wavelets. 2. Clifford Algebra. 2.1. Axioms of Clifford Algebra. 2.2. Quadratic Forms in Clifford's Geometric Algebra. 2.3. Clifford Product and Derived Products. 2.4. Determinants in Geometric Algebra. 2.5. Gram-Schmidt Orthogonalization in Geometric Algebra. 2.6. Important Clifford Geometric Algebras. 2.7. How Imaginary Numbers Become Real in Clifford Algebras. 2.8. Quaternions and Geometry of Rotations In 3, 4 Dim. 3. Geometric Calculus. 3.1. Introductory Notes on Vector Differential Calculus. 3.2. Brief Overview of Vector Differential Calculus. 3.3. Geometric Algebra for Differential Calculus. 3.4. Vector Differential Calculus. 3.5. Summary on Vector Differential Calculus. 4. Quaternion Fourier Transforms. 4.1. Fundamentals of Quaternion Fourier Transforms (QFT). 4.2. Properties of Quaternion Fourier Transform. 4.3. Special Quaternion Fourier Transforms. 4.4. From QFT To Volume-Time FT, Spacetime FT. 5. Clifford Fourier Transforms. 5.1. Overview of Clifford Fourier Transforms. 5.2. One-Sided Clifford Fourier Transforms. 5.3. Two-Sided Clifford Fourier Transforms. 5.4. Clifford Fourier Transform and Convolution. 5.5. Special Clifford Fourier Transforms. 6. Relating QFTs And CFTs. 6.1. Background. 6.2. General Geometric Fourier Transform. 6.3. CFT Due to Sommen And Bulow. 6.4. Color Image CFT. 6.5. Two-Sided CFT. 6.6. Quaternion Fourier Transform (QFT) 6.7. Quaternion Fourier Stieltjes Transform. 6.8. Quaternion Fourier Mellin Transform, Clifford Fourier Mellin Transform. 6.9. Volume-Time CFT and Spacetime CFT. 6.10. One-Sided CFTs. 6.11. Pseudoscalar Kernel CFTs. 6.12. Quaternion and Clifford Linear Canonical Transforms. 6.13. Summary Interrelationship Of QFTs, CFTs. Appendix A. Square Roots of ?1 MAPLE, Cauchy-Schwarz, Uncertainty Equality. References. Bibliography. Index.