Methods in Approximation

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.