This book summarizes important outcomes of a quarter century of developments in advanced mathematical approaches and their implementations for deconvolution. The analysis of electron and relaxation time spectra obtained from the results of appropriate physical experiments fulfilled on real samples of bulk amorphous/crystalline semiconductors and insulators as well as on nano-structured materials and devices are also discussed. The second chapter of this book depicts key features of many well-known traditional and some modern techniques for experimental investigations of electron density and time relaxation spectra in such semiconductors and insulators. as Additionally, there is an emphasis on archetypal problems related to the analysis and interpretation of the results of those experimental techniques. Some generic (though crucially important in the context of this book) physical and mathematical aspects of the polarization and relaxation processes in solids, well-known one-dimensional direct and inverse integral transforms, linear integral equations of the first and second kinds, "ill-posed" mathematical problems and specific mathematical approaches to solution(s) of those are discussed in the third, fourth and fifth chapters, respectively. A majority of the aforementioned mathematical approaches are essentially based on the so-called "regularization" concept, pioneered by famous Russian mathematicians (A. N. Tikhonov, M. M. Lavrentiev, V. K. Ivanov, V. Ya. Arsenin and their co-workers) in the second half of the twentieth century. Mathematical aspects of the regularization concept are discussed (to some extent) in the fifth chapter of the book in comparison to the similar aspects of the traditional "modelling" approach with multiple references on appropriate "original" articles and books. Thanks to distinctive features of the regularization concept, it endures a protracted history (which nowadays well exceeds 5 decades), becomes the dominant strategy for the solution of various "inverse problems", and is widely used in many types of modern applications and computational packages. In particular, the regularization algorithms are incorporated into Mathematica, Matlab, Python and Octave packages.This generic "regularization" concept had been successfully implemented by the author of this book during the development and practical realization (programming) of several essentially different regularization algorithms (described in detail in the sixth chapter of the book) for unambiguous investigations and the analysis of results of appropriated physical experiments, fulfilled over a period from 1984 to 2009, both in Russia and in Singapore. Furthermore, actual results of such experimental investigations are discussed in the seventh chapter following closely appropriate original publications, and in comparison with their counterparts obtained by traditional (e.g., "modelling") approaches. As it is also demonstrated in the seventh chapter with the relevant examples and detailed discussion(s), the implementation of the aforementioned "regularization" algorithms allows one to identify (and interpret thereafter) new important features of the intra-gap and near-band-gap electronic spectra of the amorphous and polycrystalline semiconductors and insulators. The relaxation time spectra of those materials, which are usually unattainable via the implementation of the "modelling" approach is also analyzed.It is important that the regularization concept (mathematically related to its alternative ones, e.g., the direct and inverse Radon integral transforms) has many other, very important implementations, e.g., in medical computerized tomography, security-related applications, archeology, geophysics, etc. Similar to the abovementioned spectroscopic techniques, the X-ray-based computerized tomography eventually yields vital information on features of electron density distribution in a studied object, though the desired function in the latter case rather depends on spatial variables than on energetic ones. The mathematical essence of the Radon-transform-based computerized tomography is also summarized very briefly in one of the sections of this book. A comprehensive reference list of this book comprises many other key references (for those who are interested in the aforementioned topics discussed herein).
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