TY - CONF A1 - Hitzer, Eckhard A2 - Gürlebeck, Klaus A2 - Lahmer, Tom A2 - Werner, Frank T1 - THE CLIFFORD FOURIER TRANSFORM IN REAL CLIFFORD ALGEBRAS T2 - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 04 - 06 2012, Bauhaus-University Weimar N2 - We briefly review and use the recent comprehensive research on the manifolds of square roots of −1 in real Clifford geometric algebras Cl(p,q) in order to construct the Clifford Fourier transform. Basically in the kernel of the complex Fourier transform the complex imaginary unit j is replaced by a square root of −1 in Cl(p,q). The Clifford Fourier transform (CFT) thus obtained generalizes previously known and applied CFTs, which replaced the complex imaginary unit j only by blades (usually pseudoscalars) squaring to −1. A major advantage of real Clifford algebra CFTs is their completely real geometric interpretation. We study (left and right) linearity of the CFT for constant multivector coefficients in Cl(p,q), translation (x-shift) and modulation (w -shift) properties, and signal dilations. We show an inversion theorem. We establish the CFT of vector differentials, partial derivatives, vector derivatives and spatial moments of the signal. We also derive Plancherel and Parseval identities as well as a general convolution theorem. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Computerunterstütztes Verfahren Y1 - 2012 UR - https://e-pub.uni-weimar.de/opus4/frontdoor/index/index/docId/2765 UR - https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20170314-27652 UR - http://euklid.bauing.uni-weimar.de/ikm2012 SN - 1611-4086 ER -