Heritage and Advancements in Lattice Point and Sampling Regularization Theory

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.