Approximation Theory and Applications: Piecewise Linear and Generalized Functions presents the main provisions of approximation theory, and considers existing and new methods for approximating piecewise linear and generalized functions, widely used to solve problems related to mathematical modeling of systems, processes, and phenomena in fields ranging from engineering to economics. The widespread use of piecewise linear and generalized functions is explained by the simplicity of their structure. However, challenges often arise when constructing solutions over the entire domain of these functions, requiring the use special mathematical methods to put theory into practice. This book first offers a first full foundation in approximation theory as it relates to piecewise linear and generalized functions, followed by staged methods to resolve common problems in practice, with applications examined across structural mechanics, medicine, quantum theory, signal theory, semiconductor theory, mechanical engineering, heat engineering, and other fields. Later chapters consider numerical verification of approximation methods, and approximation theory as the basis for new macroeconomic theory with impulse and jump characteristics. Each chapter includes questions for review and sample problems, accompanied by a separate Solutions Manual hosted for instructor access.
1. Introduction - p. ix 2. 1. Basic provisions and methods of approximation theory - p. 1 3. Abstract - p. 1 4. Chapter Outline - p. 1 5. 1.1 The main idea of approximation of the original function: Basic concepts - p. 1 6. 1.2 Contraction mapping principle - p. 5 7. 1.3 Weierstrass theorems on the convergence of a sequence of approximating functions - p. 6 8. 1.4 Approximation by algebraic polynomials - p. 9 9. 1.5 Approximation of piecewise linear functions by Fourier series - p. 13 10. 1.6 Fourier series for an orthogonal system of elements - p. 17 11. 1.7 Function approximation using splines - p. 19 12. 1.8 Least squares method: Linear regression - p. 22 13. 1.9 Hermite interpolation - p. 23 14. 1.10 Lebesgue functions and Lebesgue constant in polynomial interpolation - p. 25 15. 1.11 Questions for review and problems - p. 27 16. Problems - p. 27 17. 2. New methods for approximating piecewise-linear and generalized functions - p. 33 18. 3. Practical application and examples of using the developed approximation methods - p. 73 19. 4. Numerical verification of the proposed approximation methods - p. 105 20. 5. Basics of a new macroeconomic theory with impulse and jump characteristics - p. 155 21. Conclusion - p. 209 22. Appendix 1 - p. 211 23. Appendix 2 - p. 215 24. Appendix 3 - p. 218 25. Appendix 4 - p. 221 26. Appendix 5 - p. 225 27. Appendix 6 - p. 227 28. Bibliography - p. 229 29. Index - p. 243
Approximation Theory and Applications: Piecewise Linear and Generalized Functions presents the main provisions of approximation theory, and considers existing and new methods for approximating piecewise linear and generalized functions, widely used to solve problems related to mathematical modeling of systems, processes, and phenomena in fields ranging from engineering to economics. The widespread use of piecewise linear and generalized functions is explained by the simplicity of their structure. However, challenges often arise when constructing solutions over the entire domain of these functions, requiring the use special mathematical methods to put theory into practice. This book first offers a first full foundation in approximation theory as it relates to piecewise linear and generalized functions, followed by staged methods to resolve common problems in practice, with applications examined across structural mechanics, medicine, quantum theory, signal theory, semiconductor theory, mechanical engineering, heat engineering, and other fields. Later chapters consider numerical verification of approximation methods, and approximation theory as the basis for new macroeconomic theory with impulse and jump characteristics. Each chapter includes questions for review and sample problems, accompanied by a separate Solutions Manual hosted for instructor access.
1. Introduction - p. ix 2. 1. Basic provisions and methods of approximation theory - p. 1 3. Abstract - p. 1 4. Chapter Outline - p. 1 5. 1.1 The main idea of approximation of the original function: Basic concepts - p. 1 6. 1.2 Contraction mapping principle - p. 5 7. 1.3 Weierstrass theorems on the convergence of a sequence of approximating functions - p. 6 8. 1.4 Approximation by algebraic polynomials - p. 9 9. 1.5 Approximation of piecewise linear functions by Fourier series - p. 13 10. 1.6 Fourier series for an orthogonal system of elements - p. 17 11. 1.7 Function approximation using splines - p. 19 12. 1.8 Least squares method: Linear regression - p. 22 13. 1.9 Hermite interpolation - p. 23 14. 1.10 Lebesgue functions and Lebesgue constant in polynomial interpolation - p. 25 15. 1.11 Questions for review and problems - p. 27 16. Problems - p. 27 17. 2. New methods for approximating piecewise-linear and generalized functions - p. 33 18. 3. Practical application and examples of using the developed approximation methods - p. 73 19. 4. Numerical verification of the proposed approximation methods - p. 105 20. 5. Basics of a new macroeconomic theory with impulse and jump characteristics - p. 155 21. Conclusion - p. 209 22. Appendix 1 - p. 211 23. Appendix 2 - p. 215 24. Appendix 3 - p. 218 25. Appendix 4 - p. 221 26. Appendix 5 - p. 225 27. Appendix 6 - p. 227 28. Bibliography - p. 229 29. Index - p. 243
