zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A general geometric Fourier transform convolution theorem. (English) Zbl 1263.15021
Summary: The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of “A general geometric Fourier transform” by the authors [Proceedings of the 9th international conference on Clifford algebras and their applications, Bauhaus-University Weimar, Germany (2011)], covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper, we extend the former results by a convolution theorem.
15A66Clifford algebras, spinors
43A32Other transforms and operators of Fourier type
44A35Convolution (integral transforms)
[1]William Kingdon Clifford: Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics 1(4), 350–358 · doi:10.2307/2369379
[2]Eckhard Hitzer, Rafal Ablamowicz: Geometric Roots of –1 in Clifford Algebras Cl p,q with p + q 4. Advances in Applied Clifford Algebras 21(1), 121–144 (2011) · Zbl 1216.15019 · doi:10.1007/s00006-010-0240-x
[3]Eckhard M. S. Hitzer, Jacques Helmstetter and Rafal Ablamowicz, Square Roots of in Real Clifford Algebras. In K. Gürlebeck, editor, Proceedings of the 9th International Conference on Clifford Algebras and their Applications, Bauhaus-University Weimar, Germany, 2011.
[4]Bernard Jancewicz: Trivector fourier transformation and electromagnetic field. Journal of Mathematical Physics 31(8), 1847–1852 (1990) · Zbl 0711.42013 · doi:10.1063/1.528681
[5]Julia Ebling, Visualization and Analysis of Flow Fields using Clifford Convolution. PhD thesis, University of Leipzig, Germany, 2006.
[6]Eckhard Hitzer, Bahri Mawardi: Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4). Advances in Applied Clifford Algebras 18(3), 715–736 (2008) · Zbl 1177.15029 · doi:10.1007/s00006-008-0098-3
[7]Frank Sommen: Hypercomplex Fourier and Laplace Transforms I. Illinois Journal of Mathematics 26(2), 332–352 (1982)
[8]Thomas Bülow, Hypercomplex Spectral Signal Representations for Image Processing and Analysis. Inst. f. Informatik u. Prakt. Math. der Christian- Albrechts-Universität zu Kiel, 1999.
[9]Todd A. Ell, Quaternion-Fourier Transforms for Analysis of Two-Dimensional Linear Time-Invariant Partial Differential Systems. In Proceedings of the 32nd IEEE Conference on Decision and Control, volume 2, San Antonio, TX , USA, 1993, pages 1830–1841.
[10]Eckhard Hitzer: Quaternion fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras 17(3), 497–517 (2007) · Zbl 1143.42006 · doi:10.1007/s00006-007-0037-8
[11]Thomas Batard, Michel Berthier and Christophe Saint-Jean, Clifford Fourier Transform for Color Image Processing. In E. Bayro-Corrochano and G. Scheuermann, editors, Geometric Algebra Computing: In Engineering and Computer Science, Springer, London, UK 2010, pages 135–162.
[12]Fred Brackx, Nele De Schepper and Frank Sommen, The Cylindrical Fourier Transform. In E. Bayro-Corrochano and G. Scheuermann, editors, Geometric Algebra Computing: In Engineering and Computer Science, Springer, London, UK 2010, pages 107–119.
[13]Michael Felsberg, Low-Level Image Processing with the Structure Multivector. PhD thesis, University of Kiel, Germany, 2002.
[14]Todd A. Ell and Steven J. Sangwine, The Discrete Fourier Transforms of a Colour Image. Blackledge, J. M. and Turner, M. J., Image Processing II: Mathematical Methods, Algorithms and Applications 2000, 430-441.
[15]Ell T.A., Sangwine S.J.: Hypercomplex fourier transforms of color images. Image Processing, IEEE Transactions on 16(1), 22–35 (2007) · doi:10.1109/TIP.2006.884955
[16]Roxana Bujack, Gerik Scheuermann and Eckhard M. S. Hitzer, A General Geometric Fourier Transform. In K. Gürlebeck, editor, Proceedings of the 9th International Conference on Clifford Algebras and their Applications, Bauhaus- University Weimar, Germany, 2011.
[17]David Hestenes, Garret Sobczyk: Clifford Algebra to Geometric Calculus. D. Reidel Publishing Group, Dordrecht, Netherlands (1984)
[18]David Hestenes: New Foundations for Classical Mechanics. Kluwer Academic Publishers, Dordrecht, Netherlands (1986)
[19]Eckhard M. S. Hitzer, Angles Between Subspaces. In V. Skala, editor,Workshop Proceedings of Computer Graphics, Computer Vision and Mathematics, Brno University of Technology, Czech Republic, 2010. UNION Agency.
[20]Eckhard Hitzer: Angles between subspaces computed in clifford algebra. AIP Conference Proceedings 1281(1), 1476–1479 (2010)