A Course in Approximation Theory
A large portion of this book focuses on multivariable approximation theory, i.e. the approximation of functions in several variables, as opposed to the classical theory of functions in one variable. Its purpose is to guide the reader in exploring contemporary approximation theory. The textbook has 36 chapters. A central theme of the book is the problem of interpolating data by smooth multivariable functions. Several chapters investigate interesting families of functions that can be employed in this task; among them are polynomials, positive definite functions and radial basis functions. The book then moves on to the consideration of methods for concocting approximations, such as by convolutions, neural nets and interpolation at more and more points. A major departure from the theme of multivariate approximation is found in the two chapters on univariate wavelets, which comprise a significant fraction of the book.
Most of the topics in the book, heretofore accessible only through research papers, are treated from the basis of currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, the interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thin-plate splines, box splines, ridge functions and convolution. An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered throughout the book allowing the student reader to get a better understanding of the subject. The book can be used as a text for courses, seminars or even solo study and is designed for graduate students in mathematics, physics, engineering and computer science.