Complex Harmonic Splines, Periodic Quasi-Wavelets, 2000
Theory and Applications

This book, written by our distinguished colleague and friend, Professor Han-Lin Chen of the Institute of Mathematics, Academia Sinica, Beijing, presents, for the first time in book form, his extensive work on complex harmonic splines with applications to wavelet analysis and the numerical solution of boundary integral equations. Professor Chen has worked in Ap­ proximation Theory and Computational Mathematics for over forty years. His scientific contributions are rich in variety and content. Through his publications and his many excellent Ph. D. students he has taken a leader­ ship role in the development of these fields within China. This new book is yet another important addition to Professor Chen's quality research in Computational Mathematics. In the last several decades, the theory of spline functions and their ap­ plications have greatly influenced numerous fields of applied mathematics, most notably, computational mathematics, wavelet analysis and geomet­ ric modeling. Many books and monographs have been published studying real variable spline functions with a focus on their algebraic, analytic and computational properties. In contrast, this book is the first to present the theory of complex harmonic spline functions and their relation to wavelet analysis with applications to the solution of partial differential equations and boundary integral equations of the second kind. The material presented in this book is unique and interesting. It provides a detailed summary of the important research results of the author and his group and as well as others in the field.

1. Theory and Application of Complex Harmonic Spline Functions.- §1.1 The Interpolating Complex Spline Functions on ?.- §1.2 Quasi-Interpolant Complex Splines on ?.- §1.3 Complex Harmonic Splines and Their Function-theoretical Properties.- §1.4 Geometric Property of CHSF.- §1.5 Application of CHSF to Approximation of Conformal Mappings.- §1.6 Algorithm for Computing P(z).- §1.7 The Mappings Between Two Arbitrary Domains.- 2. Periodic Quasi-Wavelets.- §2.1 Periodic Orthonormal Quasi-wavelets.- §2.2 Quasi-wavelets on the Unit Circle.- §2.3 Anti-periodic Orthonormal Quasi-wavelets.- §2.4 Real Valued Periodic Quasi-wavelets.- §2.5 Other Methods in Periodic Multi-resolution Analysis.- 3. The Application of Quasi-Wavelets in Solving a Boundary Integral Equation of the Second Kind.- §3.1 Discretization.- §3.2 Simplifying the Procedure by Using PQW.- §3.3 Algorithm.- §3.4 Complexity.- §3.5 The Convergence of the Approximate Solution.- §3.6 Error Analysis.- §3.7 The Dirichlet Problem.- 4. The Periodic Cardinal Interpolatory Wavelets.- §4.1 The Periodic Cardinal Interpolatory Scaling Functions.- §4.2 The Periodic Cardinal Interpolatory Wavelets.- §4.3 Symmetry of Scaling Functions and Wavelets.- §4.4 Dual Scaling Functions and Dual Wavelets.- §4.5 Algorithms.- §4.6 Localization of PISF via Spline Approach.- §4.7 Localization of PISF via Circular Approach.- §4.8 Local Properties of PCIW.- §4.9 Examples.- Concluding Remarks.- References.- Author Index.

'...this book is a rigorous presentation of the numerous interesting mathematical properties and physical applications of complex harmonic spline functions, which is suitable not only as a reference source but also as a textbook for a special topics course or seminar. We are delighted to see the publication of this book and hope that it will foster new research and applications of complex harmonic splines and wavelets. We enthusiasticalloy recommend it to the mathematics and engineering communities.' Journal of Approximation Theory, 106 (2000)