Doctor of Philosophy (PhD)
This dissertation studies local and asymptotic properties of distributions (generalized functions) in connection to several problems in harmonic analysis, approximation theory, classical real and complex function theory, tauberian theory, summability of divergent series and integrals, and number theory. In Chapter 2 we give two new proofs of the Prime Number Theory based on ideas from asymptotic analysis on spaces of distributions. Several inverse problems in Fourier analysis and summability theory are studied in detail. Chapter 3 provides a complete characterization of point values of tempered distributions and functions in terms of a generalized pointwise Fourier inversion formula. The relation of the Fourier inversion formula with several summability procedures for divergent series and integrals is established. This work also provides formulas for jump singularities, that is, detection of edges from spectral data, which can be used as effective numerical detectors. Chapters 5 and 6 introduce new summability methods for the determination of jump discontinuities. Estimations on orders of summability are given in Chapter 8. Chapters 4 and 9 give a tauberian theory for distributional point values; this theory recovers important classical tauberians of Hardy and Littlewood, among others, for Dirichlet series. We make a complete wavelet analysis of asymptotic properties of distributions in Chapter 11. This study connects the boundary asymptotic behavior of the wavelet transform with asymptotics of tempered distributions. It is shown that our tauberian theorems become full characterizations. Chapter 10 makes a comprehensive study of asymptotic properties of distributions. Open problems in the area are solved in Chapter 10 and new tools are developed. We obtain a complete structural description of quasiasymptotics in one variable. We introduce the phi-transform for the local analysis of functions, measures, and distributions. In Chapter 7 the transform is used to study distributionally regulated functions (introduced here). Chapter 12 presents a characterization of measures in terms of the boundary behavior of this transform. We characterize the support of tempered distributions in Chapter 13 by various summability means of the Fourier transform.
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Vindas, Jasson, "Local behavior of distributions and applications" (2009). LSU Doctoral Dissertations. 1911.