The Best Approximation Method An Introduction

The most commonly used numerical techniques in solving engineering and mathematical models are the Finite Element, Finite Difference, and Boundary Element Methods. As computer capabilities continue to impro':e in speed, memory size and access speed, and lower costs, the use of more accurate but computationally expensive numerical techniques will become attractive to the practicing engineer. This book presents an introduction to a new approximation method based on a generalized Fourier series expansion of a linear operator equation. Because many engineering problems such as the multi­ dimensional Laplace and Poisson equations, the diffusion equation, and many integral equations are linear operator equations, this new approximation technique will be of interest to practicing engineers. Because a generalized Fourier series is used to develop the approxi­ mator, a "best approximation" is achieved in the "least-squares" sense; hence the name, the Best Approximation Method. This book guides the reader through several mathematics topics which are pertinent to the development of the theory employed by the Best Approximation Method. Working spaces such as metric spaces and Banach spaces are explained in readable terms. Integration theory in the Lebesque sense is covered carefully. Because the generalized Fourier series utilizes Lebesque integration concepts, the integra­ tion theory is covered through the topic of converging sequences of functions with respect to measure, in the mean (Lp), almost uniformly IV and almost everywhere. Generalized Fourier theory and linear operator theory are treated in Chapters 3 and 4.

1. Work Spaces.- 1.1. Metric Spaces.- 1.1.1. The Concept of a Metric.- 1.1.2. Metrics.- 1.1.3. Metrics Space Properties.- 1.1.4. Converging Sequences in a Metric Space.- 1.2. Linear Spaces.- 1.3. Normed Linear Spaces.- 1.4. Banach Spaces.- 1.4.1. Cauchy Sequences.- 1.4.2. Complete Normed Linear. Space (Banach Space).- 2. Integration Theory.- 2.0. Introduction.- 2.1. The Riemann and Lebesgue Integrals: Step and Simple Functions.- 2.2. Lebesque Measure.- 2.3. Measurable Functions.- 2.4. The Lebesgue Integral.- 2.4.1. Bounded Functions.- 2.4.2. Unbounded Functions.- 2.5. Key Theorems in Integration Theory.- 2.5.1. Monotone Convergence Theorem.- 2.5.2. Dominated Convergence Theorem.- 2.5.3. Egorou’s Theorem.- 2.6. Lp Spaces.- 2.6.1. m-Equivalent Functions.- 2.6.2. The Space Lp.- 2.6.3. Hölder’s Inequality.- 2.6.4. Cauchy-Bunyakouskilo-Schwarz Inequality.- 2.6.5. Minkowski’s Inequality.- 2.6.6. Triangle Inequality.- 2.7. The Metric Space, Lp.- 2.8. Convergence of Sequences.- 2.8.1. Common Modes of Convergence.- 2.8.2. Convergence in Lp.- 2.8.3. Convergence in Measure (M).- 2.8.4. Almost Uniform Convergence (AU).- 2.8.5. What Implies What?.- 2.8.6. Counterexamples.- 2.9. Capsulation.- 3: Hilbert Space and Generalized Fourier Series.- 3.0 Introduction.- 3.1. Inner Product and Hilbert Space (Finite Dimension Spaces).- 3.2. Infinite Dimension Spaces.- 3.2.1. L2 Space.- 3.2.2. Inner Product in L2 (E).- 3.2.3. Orthogonal Functions.- 3.2.4. Orthonormal Functions.- 3.3. Approximations in L2(E).- 3.3.1. Parseval’s Identify.- 3.3.2. Bessel’s Inequality.- 3.4. Vector Space Representation for Approximations: An Application.- 4. Linear Operators.- 4.0. Introduction.- 4.1. The Derivative as a Linear Operator.- 4.2. Linear Operators.- 4.3. Examples of Linear Operators in Engineering.- 4.4. Linear Operator Norms.- 5. The Best Approximation Method.- 5.0. Introduction.- 5.1. An Inner Product for the Solution of Linear Operator Equations.- 5.1.1. Definition of Inner Product and Norm.- 5.2. Orthonormalization Process.- 5.3. Generalized Fourier Series.- 5.3.1. Generalized Fourier Series.- 5.4. Approximation Error Evaluation.- 5.5. The Weighted Inner Product.- 6. The Best Approximation Method: Applications.- 6.0. Introduction.- 6.1. Sensitivity of Computational Results to Variation in the Inner Product Weighting Factor.- 6.2. Solving Two-Dimensional Potential Problems.- 6.3. Application to Other Linear Operators.- 6.4. Computer Program: Two-Dimensional Potential Problems Using Real Variable Basis Functions.- 6.4.1. Introduction.- 6.4.2. Input Data Description.- 6.4.3. Computer Program Listing.- 7. Coupling the Best Approximation and Complex Variable Boundary Element Methods.- 7.0. Introduction.- 7.1. The Complex Variable Boundary Element Method.- 7.1.1. Objectives.- 7.1.2. Definition 7.1: (Working Space, W?).- 7.1.3. Definition 7.2: (The Function ||?||).- 7.1.4. Almost Everywhere (ae) Equality.- 7.1.5. Theorem (relationship of ||?|| to ||?||2).- 7.1.6. Theorem.- 7.1.7. Theorem.- 7.2. Mathematical Development.- 7.2.1. Discussion: (A Note on Hardy Spaces).- 7.2.2. Theorem (Boundary Integral Representation).- 7.2.3. Almost Everywhere (ae) Equivalence.- 7.2.4. Theorem (Uniqueness of Zero Element in W?).- 7.2.5. Theorem (W? is a Vector Space).- 7.2.6. Theorem (Definition of the Inner-Product).- 7.2.7. Theorem (W? is on Inner-Product Space).- 7.2.8. Theorem (||?|| is a Norm on W?).- 7.2.9. Theorem.- 7.3. The CVBEM and W?.- 7.3.1. Definition 7.3: (Angle Points).- 7.3.2. Definition 7.4: (Boundary Element).- 7.3.3. Theorem.- 7.3.4. Defintion 7.5: (Linear Basis Function).- 7.3.5. Theorem.- 7.3.6. Defintion 7.6: (Global Trial Function).- 7.3.7. Theorem.- 7.3.8. Discussion.- 7.3.9. Theorem.- 7.3.10. Discussion.- 7.3.11. Theorem (Linear Independence of Nodal Expansion Functions).- 7.3.12. Discussion.- 7.3.13. Theorem.- 7.3.14. Theorem.- 7.3.15. Discussion.- 7.4. The Space W?A.- 7.4.1. Definition 7.7: (W?A).- 7.4.2. Theorem.- 7.4.3. Theorem.- 7.4.4. Discussion.- 7.4.5. Theorem.- 7.4.6. Theorem.- 7.4.7. Discussion: Another Look at W?.- 7.5. Applications.- 7.5.1. Introduction.- 7.5.2. Nodal Point Placement on ?.- 7.5.3. Flow-Field (Flow-Net) Development.- 7.5.4. Approximate Boundary Development.- 7.5.5. Applications.- 7.6. Computer Program: Two-Dimensional Potential Problems Using Analytic Basis Functions (CVBEM).- 7.6.1. Introduction.- 7.6.2. CVBEM1 Program Listing.- 7.6.3. Input Variable Description for CVBEM1.- 7.6.4. CVBEM2 Program Listing.- References.- Appendix A: Derivation of CVBEM Approximation Function.- Appendix B: Convergence of CVBEM Approximator.