Geometric Roots of -1 in Clifford Algebras Cℓp,q with p + q ≤ 4

Advances in Applied Clifford Algebras

Volumn 21, Issue 1, 2011, Pages 121-144

  Hitzer, Eckhard ^a   Abłamowicz, Rafat ^b  

^a UNIVERSITY OF FUKUI   (Japan)

^b TENNESSEE TECHNOLOGICAL UNIVERSITY   (United States)

Author keywords

Clifford (geometric) algebra; Fourier transformation; pseudo scalar; Roots of 1

Indexed keywords

EID: 79952736486     PISSN: 01887009     EISSN: 16614909     Source Type: Journal    
DOI: 10.1007/s00006-010-0240-x     Document Type: Article

References (32)

1. Sangwine S. J.: Biquaternion (Complexified Quaternion) Roots of -1. Adv. Appl. Clifford Alg. 16(1), 63-68 (2006).
2. W. K. Clifford Applications of Grassmann's Extensive Algebra. American Journal of Mathematics Pure and Applied 1 (1878), 350-358; Online: http://sinai. apphy. u-fukui. ac. jp/gcj/wkconline. html.
3. D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus. Kluwer, 1984.
4. F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis. Vol. 76 of Research Notes in Mathematics, Pitman Advanced Publishing Program, 1982.
5. T. Bülow, M. Felsberg and G. Sommer, Non-commutative Hypercomplex Fourier Transforms of Multidimensional Signals. In G. Sommer (ed.), "Geom. Comp. with Cliff. Alg., Theor. Found. and Appl. in Comp. Vision and Robotics" Springer (2001), 187-207.
6. Li C., McIntosh A., Qian T.: Clifford Algebras, Fourier Transform and Singular Convolution Operators On Lipschitz Surfaces. Revista Matematica Iberoamericana, 10(3), 665-695 (1994).
7. A. McIntosh, Clifford Algebras, Fourier Theory, Singular Integrals and Harmonic Functions on Lipschitz Domains. Chapter 1 of J. Ryan (ed.), Clifford Algebras in Analysis and Related Topics, CRC Press, Boca Raton, 1996.
8. T. Qian, Paley-Wiener Theorems and Shannon Sampling in the Clifford Analysis Setting. In R. Abłamowicz (ed.), "Clifford Algebras - Applications to Mathematics, Physics, and Engineering", Birkäuser, Basel, (2004), 115-124.
9. J. Ebling and G. Scheuermann, Clifford Fourier Transform on Vector Fields. IEEE Transactions on Visualization and Computer Graphics, 11 (4) July/August (2005), 469-479.
10. 3, 0. Adv. Appl. Clifford Alg. 16(1), 41-61 (2006).
11. n, 0, n = 3 (mod 4) based on Clifford Fourier transform. In Springer SCI book series "Applied and Numerical Harmonic Analysis" (2006), 45-54.
12. Hitzer E., Mawardi B.: Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2(mod 4) and n = 3 (mod 4). Adv. Appl. Clifford Alg. 18(3-4), 715-736 (2008).
13. Hitzer E.: Quaternion Fourier Transform on Quaternion Fields and Generalizations. Adv. Appl. Clifford Alg. 17(3), 497-517 (2007).
14. T. Bülow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. Ph. D. Thesis, Univ. of Kiel, 1999.
15. M. Felsberg, Low-Level Image Processing with the Structure Multivector. PhD thesis, Univ. of Kiel, 2002.
16. D. Hestenes, New Foundations for Classical Mechanics. Kluwer, 1999.
17. D. Hestenes, Space-Time Algebra. Gordon and Breach, 1966.
18. Li H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2009).
19. L. Dorst, The Inner Products of Geometric Algebra. In L. Dorst, C. Doran, J. Lasenby (eds.), Birkhäuser, Boston, 2002, 35-46.
20. Lounesto P.: Clifford Algebra and Spinors 2nd ed. CUP, Cambridge (2006).
21. Abłamowicz R., Fauser B., Podlaski K., Rembieliński J.: Idempotents of Clifford Algebras. Czechoslovak Journal of Physics, 53(11), 949-954 (2003).
22. R. Abłamowicz, B. Fauser, CLIFFORD - A Maple Package for Clifford Algebra Computations with Bigebra, SchurFkt, GfG - Groebner for Grassmann, Cliplus, Define, GTP, Octonion, SP, SymGroupAlgebra, and code support. http://www. math. tntech. edu/rafal/, December 2008.
23. Abłamowicz R., Oziewicz Z., Rzewuski J.: Clifford algebra approach to twistors. J. Math. Phys. 23, 231-242 (1982).
24. R. Abłamowicz, G. Sobczyk, Appendix 7. 1 of Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, Boston, 2004.
25. E. Hitzer, Geometric Calculus International - Software. http://sinai. apphy. ufukui. ac. jp/gcj/gc_int. html#software.
26. Said S., Le Bihan N., Sangwine S. J.: Fast complexified quaternion Fourier transform. IEEE Transactions on Signal Processing, 56(4), 1522-1531 (2008).
27. Mawardi B., Hitzer E.: Clifford Algebra Cl(3, 0)-valued Wavelet Transformation, Clifford Wavelet Uncertainty Inequality and Clifford Gabor Wavelets. International Journal of Wavelets, Multiresolution and Information Processing, 5(6), 997-1019 (2007).
28. E. Hitzer, Real Clifford Algebra Cl(n, 0), n = 2, 3(mod 4) Wavelet Transform. Edited by T. E. Simos et al., AIP Proceedings of ICNAAM 2009, No. 1168 (2009), 781-784.
29. E. Hitzer, Clifford (Geometric) Algebra Wavelet Transform. In V. Skala and D. Hildenbrand (eds.), Proc. of GraVisMa 2009, 02-04 Sep. 2009, Plzen, Czech Republic, 2009, 94-101. Online: http://gravisma. zcu. cz/GraVisMa-2009/Papers_2009/!_2009_GraVisMa_proceedings-FINAL. pdf.
30. B. Mawardi, E. Hitzer, S. Adji, Two-Dimensional Clifford Windowed Fourier Transform. Accepted for G. Scheuermann, E. Bayro-Corrochano (eds.), Applied Geometric Algebras in Computer Science and Engineering, Springer, New York, 2009.
31. B. Mawardi, E. Hitzer, R. Ashino, R. Vaillancourt, Windowed Fourier transform of two-dimensional quaternionic signals. Appl. Math. and Computation, 216, Iss. 8, 2366-2379, 15 June 2010.
32. Psalm 92, verse 5, New Int. Version of the Bible. www. biblegateway. com.