Fourier Series Fourier Transform and Their Applications to Mathematical Physics Instructor Solution Manual Solutions 1st Edition by Valery Serov 3319652613 9783319652610

Fourier Series Fourier Transform and Their Applications to Mathematical Physics Instructor Solution Manual Solutions 1st Edition Valery Serov – Ebook Instant Download/Delivery ISBN(s): 9783319652610,3319652613

Product details:

• ISBN 10:3319652613
• ISBN 13:9783319652610
• Author:Valery Serov

This text serves as an introduction to the modern theory of analysis and differential equations with applications in mathematical physics and engineering sciences. Having outgrown from a series of half-semester courses given at University of Oulu, this book consists of four self-contained parts. The first part, Fourier Series and the Discrete Fourier Transform, is devoted to the classical one-dimensional trigonometric Fourier series with some applications to PDEs and signal processing. The second part, Fourier Transform and Distributions, is concerned with distribution theory of L. Schwartz and its applications to the Schrödinger and magnetic Schrödinger operations. The third part, Operator Theory and Integral Equations, is devoted mostly to the self-adjoint but unbounded operators in Hilbert spaces and their applications to integral equations in such spaces.

Table contents:

1. Fourier Series and the Discrete Fourier Transform
1. Introduction
2. Formulation of Fourier Series
3. Fourier Coefficients and Their Properties
4. Convolution and Parseval’s Equality
5. Fejér Means of Fourier Series. Uniqueness of the Fourier Series.
6. The Riemann–Lebesgue Lemma
7. The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem.
8. Besov and Hölder Spaces
9. Absolute Convergence. Bernstein and Peetre Theorems.
10. Dirichlet Kernel. Pointwise and Uniform Convergence.
11. Formulation of the Discrete Fourier Transform and Its Properties.
12. Connection Between the Discrete Fourier Transform and the Fourier Transform.
13. Some Applications of the Discrete Fourier Transform.
14. Applications to Solving Some Model Equations
2. Fourier Transform and Distributions
15. Introduction
16. The Fourier Transform in Schwartz Space
17. The Fourier Transform in $$L^p(\mathbb {R}^n)$$ , $$1\le p\le 2$$
18. Tempered Distributions
19. Convolutions in S and $$S’$$
20. Sobolev Spaces
21. Homogeneous Distributions
22. Fundamental Solution of the Helmholtz Operator
23. Estimates for the Laplacian and Hamiltonian
3. Operator Theory and Integral Equations
24. Introduction
25. Inner Product Spaces and Hilbert Spaces
26. Symmetric Operators in Hilbert Spaces
27. John von Neumann’s Spectral Theorem
28. Spectra of Self-Adjoint Operators
29. Quadratic Forms. Friedrichs Extension.
30. Elliptic Differential Operators
31. Spectral Functions
32. The Schrödinger Operator
33. The Magnetic Schrödinger Operator
34. Integral Operators with Weak Singularities. Integral Equations of the First and Second Kinds.
35. Volterra and Singular Integral Equations
36. Approximate Methods
4. Partial Differential Equations
37. Introduction
38. Local Existence Theory
39. The Laplace Operator
40. The Dirichlet and Neumann Problems
41. Layer Potentials
42. Elliptic Boundary Value Problems
43. The Direct Scattering Problem for the Helmholtz Equation
44. Some Inverse Scattering Problems for the Schrödinger Operator
45. The Heat Operator
46. The Wave Operator

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