On Hypercomplex and Time-Frequency Analysis

In this thesis we study two topics in harmonic analysis. In the first half we concentrate on Clifford analysis and, in particular, derive Cauchy-type formulas for certain regular functions. In the second half of this thesis we focus on time-frequency analysis and prove characterizations of properties quadratic time-frequency transforms. Clifford analysis is a branch of mathematical analysis applying Clifford algebras to study generalizations of complex analysis. These algebras are used to construct higher dimensional analogues to complex numbers. In this context the complex analytic functions are generalized to monogenic functions which are null-solutions of certain Cauchy-Riemann or Dirac operators. Many of the results in complex analysis may be translated into higher dimensions. However, these results depend on the choice of the operator defining the family of monogenic functions. We study the theory known as modified Clifford analysis. This theory is based on the modified Cauchy-Riemann operator which is closely connected to the hyperbolic space. Working in the Poincaré upper half-space model of hyperbolic geometry, we find the k-hyperbolic harmonic fundamental solutions. Using these solutions we also prove a Cauchy-type integral formula for k-hypermonogenic functions. In the second part of this thesis we focus on time-frequency analysis. The goal of this field of study is to find representations which combine the features of both the signal and its Fourier transform. Using time-frequency representations such as time-frequency transforms signals can be described and manipulated jointly in time and in frequency. If the signal is music, a time-frequency transforms acts as its mathematical musical score. We study quadratic time-frequency transforms which may be interpreted as time-frequency energy densities of a given signal. By the Heisenberg uncertainty relation, a signal cannot be perfectly localized in time and simultaneously have a definite frequency. This precludes the existence of a perfect time-frequency energy density. Nevertheless, such an energy density may be approximated in some sense using quadratic time-frequency transforms. We consider the Cohen class of covariant time-frequency transforms and prove characterizations of several properties linked to energy densities and transformations of signals. Most of these properties are characterized in terms of the quantization, the integral kernel and the evaluation at the time-frequency origin of the given transform.