Heritage and Advancements in Lattice Point and Sampling Regularization Theory
Volume 2: From Gauss' Integral Transform to Sampling Regularization
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Beschreibung
This is the second book of two volumes dedicated to C. F. Gauss on the occasion of his 250th birthday. The objective of the two books is to demonstrate that the heritage of Gauss still has much to offer today in building a strong scientific bridge between mathematics and geoscience.
Volume 2 demonstrates that multidimensional alternating series become more tractable through convergence criteria relating oscillatory properties to an appropriate choice of a Helmholtz operator. As a consequence, new classes of lattice point identities within Shannon sampling can be developed. Wavelet sampling framework is applied to inverse antenna theory. A distinctive feature of sampling is that the coefficients governing the reproduction process for generalized measurements can be determined explicitly using methods originating from lattice point theory. Thus, Volume 2 can be regarded as a forward-looking compendium for addressing ill-posed inverse problems through sampling-based regularization.
This book will be valuable to a broad audience, including mathematicians working in number theory and applied mathematics, as well as professionals in the geosciences, such as geophysics and geoengineering.
This is the second book of two volumes dedicated to C. F. Gauss on the occasion of his 250th birthday. The objective of the two books is to demonstrate that the heritage of Gauss still has much to offer today in building a strong scientific bridge between mathematics and geoscience.
Volume 2 demonstrates that multidimensional alternating series become more tractable through convergence criteria relating oscillatory properties to an appropriate choice of a Helmholtz operator. As a consequence, new classes of lattice point identities within Shannon sampling can be developed. Wavelet sampling framework is applied to inverse antenna theory. A distinctive feature of sampling is that the coefficients governing the reproduction process for generalized measurements can be determined explicitly using methods originating from lattice point theory. Thus, Volume 2 can be regarded as a forward-looking compendium for addressing ill-posed inverse problems through sampling-based regularization.
This book will be valuable to a broad audience, including mathematicians working in number theory and applied mathematics, as well as professionals in the geosciences, such as geophysics and geoengineering.
Presents a unified framework connecting lattice point theory, sampling theory, and inverse antenna problems Develops multiscale sampling methods applicable to inverse antenna problems Introduces an inversion framework for resolving ill‑posed antenna reconstruction problems
Autor*in
Willi Freeden
Themen in »Heritage and Advancements in Lattice Point and Sampling Regularization Theory«
Lattice (Green) functions Euler- and Poisson summation formulas to Helmholtz operators Fourier integral transform Gauss-Weierstrass mollifier Shannon sampling Lattice point and ball wavelet sampling Finite region antenna problem and regularization Generalized measurements Mollifier kernel reproduction Multiscale decorrelation