Multivariate polysplines: applications to numerical and wavelet analysis
The book introduces and develops a new type of multivariate splines known as polysplines or piecewise polyharmonic splines. The main objective is the application of these tools to partial differential equations and some other areas. It is also suitable as a text book for graduate courses. The book consists of 4 chapters containing historical notes and extensive bibliography. Chapter 1 represents an introduction to polysplines. The main topics are one-dimensional and cubic splines, harmonic and polyharmonic functions in rectangular domains and annuli in \(\mathbb{R}^2\), polysplines on strips and annuli in \(\mathbb{R}^n\), spherical harmonics and polyharmonic functions, Chebyshev splines, applications to magnetism and computer aided geometric design (CAGD). Chapter 2 is devoted to cardinal polysplines in \(\mathbb{R}^n\) and covers the following topics: cardinal \(L\)-splines according to Micchelli, Riesz bounds, cardinal interpolation polysplines on annuli. Chapter 3 introduces the reader to wavelet analysis in terms of cardinal splines. Polyharmonic wavelet analysis and the relevant multiresolution analysis are developed. Chapter 4 is entitled as ``Polysplines for general interfaces''. The subtitles are heuristic argument, definition of polysplines and uniqueness for general interfaces, a priory estimates and Fredholm operators, existence and convergence of polysplines, elliptic boundary value problems in Sobolev and Hölder spaces. The book is well written. It will be helpful for researchers working in various areas of analysis, mathematical physics and computer sciences.
scientific article; zbMATH DE number 1615563