This volume studies integral transforms as tools for solving equations, analyzing signals, and transforming problems into more tractable forms.
This volume studies integral transforms as tools for solving equations, analyzing signals, and transforming problems into more tractable forms. It emphasizes both theoretical properties and computational methods.
Part I. Foundations
Chapter 1. Integral Transforms
1.1 Definitions and examples 1.2 Kernel functions 1.3 Linearity and basic properties 1.4 Inversion problems 1.5 Applications overview
Chapter 2. Operational Calculus
2.1 Algebraic manipulation of operators 2.2 Differential operators 2.3 Integral operators 2.4 Symbolic methods 2.5 Examples
Chapter 3. Convolution and Kernels
3.1 Convolution operations 3.2 Kernel representations 3.3 Convolution theorems 3.4 Applications 3.5 Examples
Part II. Classical Transforms
Chapter 4. Laplace Transform
4.1 Definition 4.2 Properties 4.3 Inversion formula 4.4 Differential equations 4.5 Applications
Chapter 5. Fourier Transform
5.1 Definition 5.2 Properties 5.3 Inversion and Plancherel theorem 5.4 Applications 5.5 Examples
Chapter 6. Mellin Transform
6.1 Definition 6.2 Properties 6.3 Inversion 6.4 Applications 6.5 Examples
Part III. Specialized Transforms
Chapter 7. Hankel and Bessel Transforms
7.1 Definitions 7.2 Properties 7.3 Applications 7.4 Examples 7.5 Connections
Chapter 8. Radon Transform
8.1 Definition 8.2 Inversion 8.3 Applications in tomography 8.4 Examples 8.5 Extensions
Chapter 9. Wavelet Transform
9.1 Continuous wavelet transform 9.2 Discrete wavelets 9.3 Multiresolution analysis 9.4 Applications 9.5 Examples
Part IV. Inversion and Uniqueness
Chapter 10. Inversion Formulas
10.1 General framework 10.2 Conditions for inversion 10.3 Uniqueness 10.4 Applications 10.5 Examples
Chapter 11. Ill-Posed Problems
11.1 Instability of inversion 11.2 Regularization methods 11.3 Applications 11.4 Examples 11.5 Techniques
Chapter 12. Asymptotic Methods
12.1 Saddle-point method 12.2 Stationary phase 12.3 Laplace method 12.4 Applications 12.5 Examples
Part V. Transform Methods for Differential Equations
Chapter 13. Solving ODEs
13.1 Laplace transform methods 13.2 Operational techniques 13.3 Initial value problems 13.4 Applications 13.5 Examples
Chapter 14. Solving PDEs
14.1 Fourier transform methods 14.2 Separation of variables 14.3 Boundary value problems 14.4 Applications 14.5 Examples
Chapter 15. Integral Equations
15.1 Transform solutions 15.2 Kernel methods 15.3 Fredholm equations 15.4 Applications 15.5 Examples
Part VI. Functional and Operator Methods
Chapter 16. Transform Operators
16.1 Operator viewpoint 16.2 Spectral representations 16.3 Functional calculus 16.4 Applications 16.5 Examples
Chapter 17. Distribution Theory
17.1 Generalized functions 17.2 Transforms of distributions 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Noncommutative Transforms
18.1 Transform methods on groups 18.2 Operator-valued transforms 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Engineering Applications
19.1 Signal processing 19.2 Control systems 19.3 Filtering 19.4 Applications 19.5 Examples
Chapter 20. Physics Applications
20.1 Quantum mechanics 20.2 Wave propagation 20.3 Heat transfer 20.4 Applications 20.5 Examples
Chapter 21. Computational Aspects
21.1 Fast transform algorithms 21.2 Numerical inversion 21.3 Stability and error 21.4 Software tools 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Modern transform methods 22.2 Multiscale analysis 22.3 Data-driven transforms 22.4 Connections to machine learning 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Inversion challenges 23.2 Stability issues 23.3 High-dimensional transforms 23.4 Computational complexity 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of transform theory 24.2 Key contributors 24.3 Evolution of operational calculus 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Transform formulas reference B. Kernel tables C. Proof techniques checklist D. Numerical methods reference E. Cross-reference to other MSC branches
This volume develops integral transforms as powerful tools for analysis and computation. It emphasizes transformation, simplification, and solution of complex problems.