Real Clifford Windowed Fourier Transform

We study the windowed Fourier transform in the framework of Clifford analysis, which\ud we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the\ud Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation,\ud reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an\ud example to show the differences between the classical windowed Fourier transform (WFT) and the\ud CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWF

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford\ud algebra Cln,0-valued admissible wavelet transform using the admissible\ud similitude group SIM(n), a subgroup of the affine group of Rn. We\ud express the admissibility condition in terms of the Cln,0 Clifford Fourier\ud transform (CFT). We show that its fundamental properties such as inner\ud product, norm relation, and inversion formula can be established\ud whenever the Clifford admissible wavelet satisfies a particular admissibility\ud condition. As an application we derive a Heisenberg type uncertainty\ud principle for the Clifford algebra Cln,0-valued admissible wavelet\ud transform. Finally, we provide some basic examples of these extended\ud wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets

Group Extensions and the Primitive Ideal Spaces of Toeplitz Algebras

Let Γ be a totally ordered abelian group and I an order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebra T(Γ) to the structure of the Toeplitz algebras T(I) and T(Γ/I). We then describe the primitive ideal space of the Toeplitz algebra T(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T(Γ) when 0→ I→ Γ→ Γ/I→ 0 is a non-trivial group extension