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Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

After developing the basics of a sampling theory and its connections to various geometric and topological properties, the author describes a suite of algorithms that have been designed for the reconstruction problem, including algorithms for surface reconstruction from dense samples, from samples that are not adequately dense and from noisy samples.

The goal of learning theory is to approximate a function from sample values. This is a general overview of the theoretical foundations, and is the first book to emphasize the approximation theory viewpoint. This emphasis provides a balanced approach, and will attract mathematicians to the problems raised.

Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models are singular: mixture models, neural networks, HMMs, and Bayesian networks are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

This book presents the basic concepts from the emerging field of computational topology that combines topology theory with the power of computing to solve problems in diverse fields. Written from a computer science perspective, the book enables non-specialists to grasp the ideas and so participate in current research in computational topology.