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The Cylindrical Fourier Transform

The Cylindrical Fourier Transform. Fred Brackx Nele De Schepper Frank Sommen. Clifford Research Group - Department of Mathematical Analysis - Ghent University. W.K. Clifford (1845 – 1879). Clifford toolbox …. CLIFFORD ALGEBRA. : orthonormal basis of. Basis of Clifford algebra.

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The Cylindrical Fourier Transform

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  1. The Cylindrical Fourier Transform Fred Brackx Nele De Schepper Frank Sommen Clifford Research Group - Department of Mathematical Analysis - Ghent University AGACSE – Leipzig, Germany August 17-19 2008

  2. W.K. Clifford (1845 – 1879) Clifford toolbox … CLIFFORD ALGEBRA :orthonormal basis of Basis of Clifford algebra or : with Identity element: Non-commutative multiplication: AGACSE – Leipzig, Germany August 17-19 2008

  3. . . . Clifford toolbox . . . Conjugation: anti-involution for which Hermitean conjugation : Hermitean inner product : Associated norm: AGACSE – Leipzig, Germany August 17-19 2008

  4. . . . Clifford toolbox . . . or • with • and In particular: AGACSE – Leipzig, Germany August 17-19 2008

  5. . . . Clifford toolbox . . . CLIFFORD ANALYSIS is left monogenic in in with : Dirac operator AGACSE – Leipzig, Germany August 17-19 2008

  6. . . . Clifford toolbox and : left solid inner spherical monogenic of order k Orthonormal basis for L2( IRm ) : thegeneralized Clifford-Hermite polynomials defined by with AGACSE – Leipzig, Germany August 17-19 2008

  7. Clifford-Fourier Transform . . . Classical tensorial Fourier transform: operator exponential form: J.-B. Joseph Fourier (1768-1830) : scalar-valued with AGACSE – Leipzig, Germany August 17-19 2008

  8. . . . Clifford-Fourier Transform . . . with : angular Dirac operator : Clifford-Fourier transform monogenicity = refinement harmonicity Clifford-Fourier transform = refinement classical Fourier transform AGACSE – Leipzig, Germany August 17-19 2008

  9. . . . Clifford-Fourier Transform multiplication rule: differentiation rule: Two - dimensional Clifford-Fourier transform: AGACSE – Leipzig, Germany August 17-19 2008

  10. Cylindrical Fourier transform – Definition . . . DEFINITION with Remark: for Expression integral kernel: with AGACSE – Leipzig, Germany August 17-19 2008

  11. . . . Cylindrical Fourier transform – Why “cylindrical”? . . . WHY “CYLINDRICAL”? For fixed, the “phase” is constant is constant or for fixed, the “phase” of the cylindrical Fourier kernel is constant on co-axial cylinders w.r.t. AGACSE – Leipzig, Germany August 17-19 2008

  12. . . . Cylindrical Fourier transform – Why “cylindrical”? . . . Comparison with classical Fourier transform: For fixed, the “phase” is constant is constant or for fixed, the level surfaces of the traditional Fourier kernel are planes perpendicular to AGACSE – Leipzig, Germany August 17-19 2008

  13. . . . Cylindrical Fourier transform – Properties . . . PROPERTIES For Differentiation rule: Multiplication rule: AGACSE – Leipzig, Germany August 17-19 2008

  14. . . . Cylindrical Fourier transform – spectrum L2- basis . . . Aim: calculate the cylindrical Fourier spectrum of the L2- basis: Calculation method is based on Funk - Hecke theorem in space: : spherical harmonic of degree k; fixed Notation: for AGACSE – Leipzig, Germany August 17-19 2008

  15. . . . Cylindrical Fourier transform – spectrum L2- basis . . . A) The cylindrical Fourier spectrum of 1) k even with : generalized hypergeometric series : Pochhammer’s symbol AGACSE – Leipzig, Germany August 17-19 2008

  16. . . . Cylindrical Fourier transform – spectrum L2- basis . . . with : Kummer’s function AGACSE – Leipzig, Germany August 17-19 2008

  17. . . . Cylindrical Fourier transform – spectrum L2- basis . . . Special case: p=0 for m= 3, 4, 5, 6 AGACSE – Leipzig, Germany August 17-19 2008

  18. . . . Cylindrical Fourier transform – spectrum L2- basis . . . 2) k odd Special case: p=0 AGACSE – Leipzig, Germany August 17-19 2008

  19. . . . Cylindrical Fourier transform – spectrum L2- basis . . . B) The cylindrical Fourier spectrum of 1) k even for m= 3, 4, 5, 6 AGACSE – Leipzig, Germany August 17-19 2008

  20. . . . Cylindrical Fourier transform – spectrum L2- basis . . . 2) k odd AGACSE – Leipzig, Germany August 17-19 2008

  21. . . . Cylindrical Fourier transform – example . . . Characteristic function of a geodesic triangle on S2 : Computation by means of spherical co-ordinates: with AGACSE – Leipzig, Germany August 17-19 2008

  22. . . . Cylindrical transform – example . . . Results simulation (Maple): real part of e1 e2 – component of AGACSE – Leipzig, Germany August 17-19 2008

  23. . . . Cylindrical transform – example Results simulation (Maple): e2 e3 – component of e1 e3 – component of AGACSE – Leipzig, Germany August 17-19 2008

  24. References • F. Brackx, N. De Schepper and F. Sommen, The Clifford-Fourier Transform, • J. Fourier Anal. Appl. 11(6) (2005), 669 – 681. • F. Brackx, N. De Schepper and F. Sommen, The Two-Dimensional Clifford- • Fourier Transform, J. Math. Imaging Vision 26(1-2) (2006), 5 – 18. • F. Brackx, N. De Schepper and F. Sommen, The Fourier Transform in Clifford • Analysis, to appear in Advances in Imaging & Electron Physics. AGACSE – Leipzig, Germany August 17-19 2008

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