Methods in Approximation

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Contenu

Basic Concepts.- Integral Domains, Fields and Vector Spaces.- Subspaces, Bases and Inner Products.- Spaces, Subspaces and Approximation.- The Continuous Function.- Polynomial Subspaces.- Spaces Generated by Differential Equations.- The Piecewise Linear Function.- Discussion.- Bibliograpy and Comments.- Polynomial Approximation.- Piecewise Linear Functions.- Curve Fitting by Straight Lines.- A One Dimensional Process in Dynamic Programming.- The Functional Equation.- The Principle of Optimality.- A Direct Derivation.- Curve Fitting by Segmented Straight Lines.- A Dynamic Programming Approach.- A Computational Procedure.- Three Dimensional Polygonal Approximation.- The Orthogonal Polynomials.- The Approximation Technique.- Discussion.- Bibliography and Comments.- Polynomial Splines.- The Cubic Spline I.- Construction of the Cubic Spline.- Existence and Uniqueness.- A Computational Algorithm - Potter's Method.- Splines via Dynamic Programming.- Derivation of Splines by Dynamic Programming.- Equivalence of the Recursive Relations obtained by Dynamic Programming and the Usual results.- Cardinal Splines.- Polynomial Splines.- Generalized Splines.- Mean Square Spline Approximation.- The Cubic Spline II.- The Minimization Procedure.- The Functional Equation.- Recursion Relations.- Bibliography and Comments.- Quasilinearization.- Quasilinearization I.- The Newton Raphson Method.- Quasilinearization II.- Existence.- Convergence.- An Example, Parameter Identification.- Unknown Initial Conditions.- Damped Oscillations.- Segmental Differential Approximation.- Differential Systems with Time Varying Coefficients.- A Method of Solution.- An Interesting Case.- Discussion.- Bibliography and Comments.- Differential Approximation.- Differential Approximation.- Linear Differential Operators.- Degree of Approximation.- Improving the Approximation.- An Example.- Differential-Difference Equations.- A Useful Approximation to g(t).- Discussion.- An Example.- Functional Differential Equations.- The Nonlinear Spring.- The Van der Pol Equation.- Bibliography and Comments.- Differential Quadrature.- Differential Quadrature.- Determination Of the Weighting Coefficients.- A First Order Problem.- A Nonlinear Wave Equation.- Systems of Nonlinear Partial Differential Equations.- Higher Order Systems.- Long Term Integration.- G(y) Linear.- G(y) Nonlinear.- A Mathematical Problem.- Systems with Partial Information.- Bibliography and Comments.- Exponential Approximation.- Approximation in Function Space.- An Example - Pharmacokinetics.- Other Physical Processes.- Proney's Method.- The Renewal Equation.- The Fredholm Integral Equation.- Bibliography and Comments.- The Riccati Equation.- The Linear Differential Equation.- Differential Inequalities.- Solution of the Riccati Equation in terms of the Maximum Operation.- Upper and Lower Bounds.- Successive Approximations via Quasilinearization.- An Illustrative Example.- Higher Order Approximations.- Multidimensional Riccati Equation.- Variational Problems and the Riccati Equation.- Bibliography and Comments.- Solution of Approximate Equations.- First Order Differential Equations.- The Second Order Differential Equation.- Discussion.- Linear Perturbations.- The Van der Pol Equation I.- The Van der Pol Equation II.- The Riccati Equation.- u? + a(t)u = 0.- Another Approach.- Discussion.- Bibliography and Comments.- Magnetic Field Determination.- The Theoretical Problem.- Maxwell's Equations.- A Variational Principle.- The Finite Element Method.- Computational Aspects.- Analytical Considerations.- Boundary Conditions.- Discussion.- Bibliography and Comments.