Fourier and wavelet analysis of Clifford-valued functions

Fourier analysis has long been studied as a method to analyse real-valued or complex-valued signals. The Clifford-Fourier transform recently developed by Brackx, De Schepper, and Sommen has led to the development of Fourier analytic methods for hypercomplex or Clifford-valued signals. In the quaternionic case, Brackx et al. have found the kernel of the Quaternionic Fourier transform which allows for much easier calculation, and we focus much of our attention in this thesis on the quaternionic case. We define the continuous wavelet transform of quaternion-valued signals on the plane and prove a Calderón reproducing formula. We also define the monogenic signal, a generalization of the analytic signal of a function on the real line. We provide a characterization of translation-invariant operators and submodules of the quaternionic L₂ module. We develop several fundamental analogues of classical orthogonal wavelet theory pioneered by Cohen, Daubechies, Mallat, and Meyer to quaternion-valued functions on the plane. We include design conditions required to produce wavelets which have compact support and desired regularity. We also develop the basic theory needed for constructing a biorthogonal wavelet basis and construct an example. For a general Clifford algebra, we develop a condition on f so that f*g satisfies a convolution theorem. We also develop a Clifford-Fourier characterization of the Clifford-valued Hardy spaces on ℝ^d.