Time-resolved fluorescence spectroscopy (TRFS) is a powerful analytical tool for quantifying the biochemical composition of organic and inorganic materials. The potentials of TRFS as nondestructive clinical tool for tissue diagnosis have been recently demonstrated. To facilitate the translation of TRFS technology to the clinical arena, algorithms for online TRFS data analysis are of great need. A fast model-free TRFS deconvolution algorithm based on the Laguerre expansion method has been previously introduced, demonstrating faster performance than standard multiexponential methods, and the ability to estimate complex fluorescence decay without any a-priori assumption of its functional form. One limitation of this method, however, was the need to select, a priori, the Laguerre parameter a and the expansion order, which are crucial for accurate estimation of the fluorescence decay. In this thesis, a new implementation of the Laguerre deconvolution method is introduced, in which a nonlinear least-square optimization of the Laguerre parameter is performed, and the selection of optimal expansion order is attained based on a Minimum Description Length (MDL) criterion. In addition, estimation of the zero-time delay between the recorded instrument response and fluorescence decay is also performed based on a normalized means square error criterion. The method was fully validated on fluorescence lifetime, endogenous tissue fluorophores, and human tissue. The automated Laguerre deconvolution method is expected to facilitate online applications of TRFS, such as clinical real-time tissue diagnosis.

We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions. Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.