Approximation Theory and Applications
Piecewise Linear and Generalized Functions
- 1st Edition - February 4, 2025
- Imprint: Academic Press
- Author: Sergei Aliukov
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 2 9 1 4 1 - 8
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 2 9 1 4 2 - 5
Approximation Theory and Applications: Piecewise Linear and Generalized Functions presents the main provisions of approximation theory, and considers existing and new method… Read more
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Request a sales quoteApproximation 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.
- Offers clear, comprehensive coverage of approximation theory and applications, with full consideration for newly evolved implications of piecewise linear and generalized functions
- Features practical examples across structural mechanics, medicine, quantum theory, signal theory, semiconductor theory, mechanical engineering, and heat engineering, among other fields
- Includes questions for review, sample problems, and a separate Solutions Manual hosted for instructor access
- Considers numerical verification of approximation methods
Undergraduate students in mathematics, statistics, business, economics, engineering and related STEM and non-STEM majors
- Title of Book
- Cover image
- Title page
- Table of Contents
- Copyright
- About the Author
- Introduction
- 1. Basic provisions and methods of approximation theory
- Abstract
- Chapter Outline
- 1.1 The main idea of approximation of the original function: Basic concepts
- 1.2 Contraction mapping principle
- 1.3 Weierstrass theorems on the convergence of a sequence of approximating functions
- 1.4 Approximation by algebraic polynomials
- 1.5 Approximation of piecewise linear functions by Fourier series
- 1.6 Fourier series for an orthogonal system of elements
- 1.7 Function approximation using splines
- 1.8 Least squares method: Linear regression
- 1.9 Hermite interpolation
- 1.10 Lebesgue functions and Lebesgue constant in polynomial interpolation
- 1.11 Questions for review and problems
- Problems
- 2. New methods for approximating piecewise-linear and generalized functions
- Abstract
- Chapter Outline
- 2.1 Disadvantages of approximating piecewise linear functions by the Fourier series
- 2.2 Description of new methods of approximation of piecewise-linear functions and their convergence
- 2.3 Approximation error
- 2.4 Generalized functions and their approximation by a sequence of recursive functions
- 2.5 Approximation of derivatives of generalized functions: Comparison of approximation methods
- 2.6 Approximation of functions in jump processes with a sharp change in the state of the research object
- 2.7 Questions for review and problems
- Problems
- 3. Practical application and examples of using the developed approximation methods
- Abstract
- Chapter Outline
- 3.1 Examples of approximation of step and impulse functions
- 3.2 Examples of approximation of piecewise linear functions of general view
- 3.3 Application of new methods of approximation in problems of structural mechanics
- 3.4 Application of new approximation methods for modeling diffusion processes in semiconductor materials
- 3.5 Examples of practical application of approximation methods in various fields of research
- 3.6 Questions for review and problems
- Problems
- 4. Numerical verification of the proposed approximation methods
- Abstract
- Chapter Outline
- 4.1 Numerical verification on the example of the dynamicsof inertial continuously variable transmissions
- 4.2 Numerical comparative analysis of methods for approximating step functions using the example of the dynamics of the freewheel mechanism
- 4.3 Numerical verification of the proposed methods for approximating step functions using the example of a nonlinear characteristic of an adaptive suspension of a vehicle
- 4.4 Approximation of nonlinear characteristics of dynamicsystems using the example of adaptive vehicle suspension
- 4.5 Eliminating the negative Gibbs effect
- 4.6 Questions for review and problems
- Problems
- 5. Basics of a new macroeconomic theory with impulse and jump characteristics
- Abstract
- Chapter Outline
- 5.1 Current problems of macroeconomics
- 5.2 Main types of impulse and step characteristics
- 5.3 The system of macroeconomic indicators
- 5.4 Approaches to methods for forecasting macroeconomic events
- 5.5 Comparative multidimensional analysis of the current state of European economies based on the complex of macroeconomic indicators
- 5.6 Chapter conclusions
- 5.7 Questions for review and problems
- Problems
- Conclusion
- Appendix 1
- Approximation of the Heaviside function, δ-function, and its derivatives
- Appendix 2
- An example of a computer program for constructing a toothed profile using the proposed approximating procedure
- Appendix 3
- An example of a computer program for numerical verification of the proposed methods of approximation of step functions based on the dynamics of inertial stepless transmission
- Appendix 4
- An example of a computer program for the numerical verification of approximation methods based on the dynamics of an inertial continuously variable transmission without freewheeling mechanisms
- Appendix 5
- An example of a computer program for the numerical comparison of methods for approximating step functions based on the dynamics of the relay freewheeling mechanism
- Appendix 6
- Numerical verification of approximation methods based on the example of studying the w of the stiffness of the car’s suspension
- Bibliography
- Index
- Edition: 1
- Published: February 4, 2025
- Imprint: Academic Press
- No. of pages: 304
- Language: English
- Paperback ISBN: 9780443291418
- eBook ISBN: 9780443291425
SA
Sergei Aliukov
Sergei Aliukov is a Professor at South Ural State University, Chelyabinsk, Russia. He received his Ph.D. degree in mechanical engineering from South Ural State University in 1983 and the Doctor's degree in the field of mechanical engineering from the South Ural State University in 2005. He has more than 150 publications, and 10 inventions to his name, and his research interests include mathematical modeling processes and applications and non-linear dynamic systems. He is a Guest Editor of the Special Issue of the journal Mathematics.
Affiliations and expertise
Professor, South Ural State University, Chelyabinsk, RussiaRead Approximation Theory and Applications on ScienceDirect