Spherical sampling
The authors have many years of experience in the field of geomathematics and now present a mathematically demanding textbook on spherically reflected sampling under geomathematically significant aspects. From the Preface: The major purpose is to give a consistent and unified overview of diverse results and various developments in the field of today's spherical sampling, particularly those arising in mathematical geosciences such as mathematical geodesy and geophysics. Although the book often refers to original contributions, we have tried to make it accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering interesed in spherical sampling theory and to show how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches, for example, neuro-medicine. From the Introduction: The special interest in writing this monograph is to specify and classify general sampling types of polynomial (i.e. spherical harmonics) expansion, spline interpolation, and wavelet approximation on the unit sphere corresponding to structured and scattered data. The large class of possible weights that are permitted in bandlimited polynomial as well as non-bandlimited spline sampling should allow the users to built up some desirable characterisrics for their own specific application under consideration. [\dots] This book is written under the seminal auspices that there is no universal method in spherical sampling, being optimal in all aspects. The book presents two essential approaches, namely plane involved sampling (Chapters 6 to 10) and sphere intrinsic sampling (Chapters 11 to 16). The first one fills out the first half of the book and uses tools from a transfer back to the bivariate Euclidean space such as stereographic mapping, Kelvin transform, polar coordinates. The second one takes advantage of the rotational invariance on the sphere and the rotational-invariant character of pseudodifferential operators representing geoscientifically relevant observables. The whole book is subdivided into ten parts with nineteen chapters: 1. Introduction 2. Basics and Settings 3. Spherical Harmonics 4. Zonal Functions 5. Slepian Functions: Basics and Settings 6. Stereographic Shannon-Type Sampling 7. Plane Based Scaling and Wavelet Functions 8. Sampling Based on Bivariate Fourier Coefficient Integration 9. Orthogonal Zonal, Tesseral, and Sectorial Wavelet Reconstruction 10. Biorthogonal Finite-Cap-Element Multiscale Tree Sampling 11. Spherical Harmonics Interpolatory Sampling 12. Bandlimited Multiscale Tree Sampling 13. RKHS Framework and Spline Sampling 14. Orthogonal/Non-Orthogonal Wavelet Approximations and Tree Sampling 15. Non-Orthogonal Finite-Cap-Element Multiscale Sampling 16. Non-Orthogonal Up Function Multiscale Tree Sampling 17. Sampling Solutions of Inverse Pseudodifferential Equations 18. Sampling of Potential and Stream Functions 19. Applicabilities and Applications.
scientific article; zbMATH DE number 6863919