Fourier Series.
Bhatia, Rejendra.
Fourier Series. - 1st ed. - 1 online resource (131 pages)
Intro -- copyright page -- title page -- Contents -- Preface -- 0 A History of Fourier Series -- 1. The motion of a vibrating string -- 2. J. D'Alembert -- 3. L. Euler -- 4. D. Bernoulli -- 5. J. Fourier -- 6. P. Dirichlet -- 7. B. Riemann -- 8. P. du Bois-Reymond -- 9. G. Cantor -- 10. L. Fejér -- 11. H. Lebesgue -- 12. A.N. Kolmogorov -- 13. L. Carleson -- 14. The L_2 theory and Hilbert spaces -- 15. Some modern developments-I -- 16. Some modern developments-II -- 17. Pure and applied mathematics -- Chapt 1 Heat Conduction and Fourier Series -- 1.1 The Laplace equation in two dimensions -- 1.2 Solutions of the Laplace equation -- 1.3 The complete solution of the Laplace equation -- 2 Convergence of Fourier Series -- 2.1 Abel summability and Cesàro summability -- 2.2 The Dirichlet and the Fejér kernels -- 2.3 Pointwise convergence of Fourier series -- 2.4 Term by term integration and differentiation -- 2.5 Divergence of Fourier series -- 3 Odds and Ends -- 3.1 Sine and cosine series -- 3.2 Functions with arbitrary periods -- 3.3 Some simple examples -- 3.4 Infinite products -- 3.5 π and infinite series -- 3.6 Bernoulli numbers -- 3.7 sinx/x -- 3.8 The Gibbs phenomenon -- 3.9 Exercises -- 3.10 A historical digression -- 4 Convergence in L_2 and L_1 -- 4.1 L_2 convergence of Fourier series -- 4.2 Fourier coefficients of L_1 functions -- 5 Some Applications -- 5.1 An ergodic theorem and number theory -- 5.2 The isoperimetric problem -- 5.3 The vibrating string -- 5.4 Band matrices -- A A Note on Normalisation -- B A Brief Bibliography -- Analysis -- Fourier series -- General reading -- History and biography -- Index -- Notation -- About the Author.
This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn the subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis.The book begins with a brief summary of the rich history of Fourier series over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory then provides unforeseen applications in diverse areas.The author starts out with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are then discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides materials at a more advanced level suitable for graduate students. The author demonstrates applications of the theory to a broad range of problems.The exercises of varying levels of difficulty that are scattered throughout the book will help readers test their understanding of the material.
9781614441045
Fourier series.
Electronic books.
QA404.B48 2005eb
515.2433
Fourier Series. - 1st ed. - 1 online resource (131 pages)
Intro -- copyright page -- title page -- Contents -- Preface -- 0 A History of Fourier Series -- 1. The motion of a vibrating string -- 2. J. D'Alembert -- 3. L. Euler -- 4. D. Bernoulli -- 5. J. Fourier -- 6. P. Dirichlet -- 7. B. Riemann -- 8. P. du Bois-Reymond -- 9. G. Cantor -- 10. L. Fejér -- 11. H. Lebesgue -- 12. A.N. Kolmogorov -- 13. L. Carleson -- 14. The L_2 theory and Hilbert spaces -- 15. Some modern developments-I -- 16. Some modern developments-II -- 17. Pure and applied mathematics -- Chapt 1 Heat Conduction and Fourier Series -- 1.1 The Laplace equation in two dimensions -- 1.2 Solutions of the Laplace equation -- 1.3 The complete solution of the Laplace equation -- 2 Convergence of Fourier Series -- 2.1 Abel summability and Cesàro summability -- 2.2 The Dirichlet and the Fejér kernels -- 2.3 Pointwise convergence of Fourier series -- 2.4 Term by term integration and differentiation -- 2.5 Divergence of Fourier series -- 3 Odds and Ends -- 3.1 Sine and cosine series -- 3.2 Functions with arbitrary periods -- 3.3 Some simple examples -- 3.4 Infinite products -- 3.5 π and infinite series -- 3.6 Bernoulli numbers -- 3.7 sinx/x -- 3.8 The Gibbs phenomenon -- 3.9 Exercises -- 3.10 A historical digression -- 4 Convergence in L_2 and L_1 -- 4.1 L_2 convergence of Fourier series -- 4.2 Fourier coefficients of L_1 functions -- 5 Some Applications -- 5.1 An ergodic theorem and number theory -- 5.2 The isoperimetric problem -- 5.3 The vibrating string -- 5.4 Band matrices -- A A Note on Normalisation -- B A Brief Bibliography -- Analysis -- Fourier series -- General reading -- History and biography -- Index -- Notation -- About the Author.
This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn the subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis.The book begins with a brief summary of the rich history of Fourier series over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory then provides unforeseen applications in diverse areas.The author starts out with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are then discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides materials at a more advanced level suitable for graduate students. The author demonstrates applications of the theory to a broad range of problems.The exercises of varying levels of difficulty that are scattered throughout the book will help readers test their understanding of the material.
9781614441045
Fourier series.
Electronic books.
QA404.B48 2005eb
515.2433