Multivariate Splines

The subject of multivariate splines has become a rapidly growing field of mathematical research. The author presents the subject from an elementary point of view that parallels the theory and development of univariate spline analysis. To compensate for the missing proofs and details, an extensive bibliography has been included. There is a presentation of open problems with an emphasis on the theory and applications to computer-aided design, data analysis, and surface fitting. Applied mathematicians and engineers working in the areas of curve fitting, finite element methods, computer-aided geometric design, signal processing, mathematical modelling, computer-aided design, computer-aided manufacturing, and circuits and systems will find this monograph essential to their research.

Univariate Splines: B-splines and truncated powers on uniform mesh; Univariate spline spaces; Some basic properties of B-splines; B-spline series; Computation of B-splines; Box Splines and Multivariate Truncated Powers: Box splines; Basic properties of box splines; Multivariate truncated powers; Box spline series; Bivariate Splines on Three and Four Directional Meshes: Dimension; Locally supported splines; Minimal and quasi-minimal supported bivariate splines; Bases and approximation order; Quasi-Interpolation Schemes: The commutator operator; Polynomial-generating formulas; Construction of quasi-interpolants; Neumann series approach; Multivariate Interpolation: Interpolation by polynomials; Lagrange interpolation by multivariate splines; Cardinal interpolation with nonsingular; Cardinal interpolation with singular; Scaled cardinal interpolation; Shape-Preserving Approximation and Other Applications: Shape-preserving approximation by box spline series; Shape-preserving quasi-interpolation and interpolation; Application of CAGD; Reconstruction of gradient fields; Applications to signal processing.

The subject of multivariate splines has become a rapidly growing field. The author presents the subject from an elementary point of view that parallels the theory and development of univariate spline analysis. Includes many open problems with emphasis on thetheory and applications to computer-aided design, data analysis, and surface fitting.