Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

1. Front Matter

   Pages i-xi

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2. Fourier Series and the Discrete Fourier Transform

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • Valery Serov

      Pages 3-10

   3. Formulation of Fourier Series

      • Valery Serov

      Pages 11-15

   4. Fourier Coefficients and Their Properties

      • Valery Serov

      Pages 17-22

   5. Convolution and Parseval’s Equality

      • Valery Serov

      Pages 23-25

   6. Fejér Means of Fourier Series. Uniqueness of the Fourier Series.

      • Valery Serov

      Pages 27-31

   7. The Riemann–Lebesgue Lemma

      • Valery Serov

      Pages 33-35

   8. The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem.

      • Valery Serov

      Pages 37-44

   9. Besov and Hölder Spaces

      • Valery Serov

      Pages 45-51

   10. Absolute Convergence. Bernstein and Peetre Theorems.

       • Valery Serov

       Pages 53-58

   11. Dirichlet Kernel. Pointwise and Uniform Convergence.

       • Valery Serov

       Pages 59-75

   12. Formulation of the Discrete Fourier Transform and Its Properties.

       • Valery Serov

       Pages 77-84

   13. Connection Between the Discrete Fourier Transform and the Fourier Transform.

       • Valery Serov

       Pages 85-92

   14. Some Applications of the Discrete Fourier Transform.

       • Valery Serov

       Pages 93-98

   15. Applications to Solving Some Model Equations

       • Valery Serov

       Pages 99-128

3. Fourier Transform and Distributions

   1. Front Matter

      Pages 129-129

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   2. Introduction

      • Valery Serov

      Pages 131-132

   3. The Fourier Transform in Schwartz Space

      • Valery Serov

      Pages 133-141

   4. The Fourier Transform in \(L^p(\mathbb {R}^n)\), \(1\le p\le 2\)

      • Valery Serov

      Pages 143-152

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. The fourth and final part, Introduction to Partial Differential Equations, servesas an introduction to modern methods for classical theory of partial differential equations.

Complete with nearly 250 exercises throughout, this text is intended for graduate level students and researchers in the mathematical sciences and engineering. 

• Fourier Series
• Fourier Transforms
• Operator Theory
• Mathematical Phsycis
• Partial Differential Equations
• Freidrich's Extension
• Schrodinger Operator
• Scattering Theory
• Sobolev Spaces
• Born Approximation
• Elliptic Operator
• Fundamental Solution
• Reisz Theory

“This book is an introduction to the modern theory of harmonic analysis and differential equations. … the book would be useful for graduate students as well as for researchers who look for applications in mathematical physics and engineering sciences.” (Sergeĭ Yu Tikhonov, Mathematical Reviews, February, 2019)

• Department of Mathematical Sciences, University of Oulu, Oulu, Finland

  Valery Serov

Valery Serov is Professor of Mathematics at the University of Oulu. Professor Serov received his PhD in Applied Mathematics in 1979 from Lomonosov Moscow State University.  He has over 120 publications, including 3 textbooks published in Russian.

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