Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4)

Abstract

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we define a generalized real Fourier transform on Clifford multivector-valued functions (f : R n → Cln,0, n = 2, 3 (mod 4)). Third, we show a set of important properties of the Clifford Fourier transform on Cln,0, n = 2, 3 (mod 4) such as differentiation properties, and the Plancherel theorem, independent of special commutation properties. Fourth, we develop and utilize commutation properties for giving explicit formulas for f x m , f ∇ m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cln,0, n = 2, 3 (mod 4) multivector functions. Mathematics Subject Classification (2000). 15A66, 43A32.

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

“…In Cℓ 2 this includes, for b 1 = b 2 = 0, the solution A = ±e 12 , which also appears in [20] on page 29. …”

“…2 In order to avoid a discussion of deviating definitions of the inner product for r = 0 or s = 0, we exclude scalars in (12), but depending on the definition of the inner product (or contraction), a single general formula for all grades exists. For example for the left contraction [19] A, B ∈ Cℓp,q, A⌋B = P r,s A r B s s−r we have the two formulas (A ∧ B)In = A⌋(BIn) and (A⌋B)In = A ∧ (BIn).…”

Quaternion and Clifford Fourier Transforms and Wavelets

“…e n squares to −1, but it ceases to be central. So the relative order of the factors in F n {f }(ω) becomes important, see [66] for a systematic investigation and comparison. In the context of generalizing quaternion Fourier transforms (QFT) via algebra isomorphisms to higher dimensional Clifford algebras, Hitzer [54] constructed a spacetime Fourier transform (SFT) in the full algebra of spacetime C 3,1 , which includes the CFT (2.1) as a partial transform of space.…”

A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

“…The generalization uses the kernel of the Cl n,0 Clifford Fourier transform [15]. It is shown that many WFT properties still hold but others have to be modified.…”