Polynomial interpolation

Abstract: 1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5. Two Classical Approximation Results.- 1.6. Exercises.- 1.7. Notes and Comment.- 2. General Properties of Interpolation Spaces.- 2.1. Categories and Functors.- 2.2. Normed Vector Spaces.- 2.3. Couples of Spaces.- 2.4. Definition of Interpolation Spaces.- 2.5. The Aronszajn-Gagliardo Theorem.- 2.6. A Necessary Condition for Interpolation.- 2.7. A Duality Theorem.- 2.8. Exercises.- 2.9. Notes and Comment.- 3. The Real Interpolation Method.- 3.1. The K-Method.- 3.2. The J-Method.- 3.3. The Equivalence Theorem.- 3.4. Simple Properties of ??, q.- 3.5. The Reiteration Theorem.- 3.6. A Formula for the K-Functional.- 3.7. The Duality Theorem.- 3.8. A Compactness Theorem.- 3.9. An Extremal Property of the Real Method.- 3.10. Quasi-Normed Abelian Groups.- 3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups.- 3.12. Some Other Equivalent Real Interpolation Methods.- 3.13. Exercises.- 3.14. Notes and Comment.- 4. The Complex Interpolation Method.- 4.1. Definition of the Complex Method.- 4.2. Simple Properties of ?[?].- 4.3. The Equivalence Theorem.- 4.4. Multilinear Interpolation.- 4.5. The Duality Theorem.- 4.6. The Reiteration Theorem.- 4.7. On the Connection with the Real Method.- 4.8. Exercises.- 4.9. Notes and Comment.- 5. Interpolation of Lp-Spaces.- 5.1. Interpolation of Lp-Spaces: the Complex Method.- 5.2. Interpolation of Lp-Spaces: the Real Method.- 5.3. Interpolation of Lorentz Spaces.- 5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1.- 5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1.- 5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences.- 5.7. Exercises.- 5.8. Notes and Comment.- 6. Interpolation of Sobolev and Besov Spaces.- 6.1. Fourier Multipliers.- 6.2. Definition of the Sobolev and Besov Spaces.- 6.3. The Homogeneous Sobolev and Besov Spaces.- 6.4. Interpolation of Sobolev and Besov Spaces.- 6.5. An Embedding Theorem.- 6.6. A Trace Theorem.- 6.7. Interpolation of Semi-Groups of Operators.- 6.8. Exercises.- 6.9. Notes and Comment.- 7. Applications to Approximation Theory.- 7.1. Approximation Spaces.- 7.2. Approximation of Functions.- 7.3. Approximation of Operators.- 7.4. Approximation by Difference Operators.- 7.5. Exercises.- 7.6. Notes and Comment.- References.- List of Symbols.

Abstract: 1. Applications and motivations 2. Hear spaces and multivariate polynomials 3. Local polynomial reproduction 4. Moving least squares 5. Auxiliary tools from analysis and measure theory 6. Positive definite functions 7. Completely monotine functions 8. Conditionally positive definite functions 9. Compactly supported functions 10. Native spaces 11. Error estimates for radial basis function interpolation 12. Stability 13. Optimal recovery 14. Data structures 15. Numerical methods 16. Generalised interpolation 17. Interpolation on spheres and other manifolds.

Abstract: From the Publisher: "In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent." This is the first book devoted to the subject and the author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

Abstract: An iterative algorithm is proposed for nonlinearly constrained optimization calculations when there are no derivatives. Each iteration forms linear approximations to the objective and constraint functions by interpolation at the vertices of a simplex and a trust region bound restricts each change to the variables. Thus a new vector of variables is calculated, which may replace one of the current vertices, either to improve the shape of the simplex or because it is the best vector that has been found so far, according to a merit function that gives attention to the greatest constraint violation. The trust region radius ρ is never increased, and it is reduced when the approximations of a well-conditioned simplex fail to yield an improvement to the variables, until ρ reaches a prescribed value that controls the final accuracy. Some convergence properties and several numerical results are given, but there are no more than 9 variables in these calculations because linear approximations can be highly inefficient. Nevertheless, the algorithm is easy to use for small numbers of variables.