A Primer on Fourier Series

Serena Dipierro is a Professor of Mathematics, Fellow of the Australian Mathematical Society, and Australian Mathematical Society Hanna Neumann Lecturer. She has obtained prestigious awards, including the Australian Mathematical Society Medal, the Mahony-Neumann-Room Prize, the Bartolozzi Prize of the Italian Mathematical Union, the Christopher Heyde Medal of the Australian Academy of Science, the Book Prize of the Italian Mathematical Union, and the Prize of the Bulletin of the Brazilian Mathematical Society.David Pfefferlé is a Swiss-Australian mathematician and plasma physicist whose career has taken him from Lausanne to Rokkasho, from Oxfordshire to Princeton, and finally to Perth, where he is a Senior Lecturer at the University of Western Australia. His research spans plasma physics, numerical modelling, and the geometry of physical systems.Enrico Valdinoci is a Professor of Mathematics and Australian Laureate Fellow. He is a highly cited researcher, Ladyzhenskaya Lecturer and has been awarded the James S. W. Wong Prize, the Mahony-Neumann-Room Prize, the Orazio Arena Prize, the Book Prize of the Unione Matematica Italiana, the Prize of the Bulletin of the Brazilian Mathematical Society, and the Amerio Gold Medal Prize.

• Where to start - 1: Fourier Series: Development of the Theory 1.1 Periodic functions 1.2 How to "optimally" approximate periodic functions 1.3 Some further observations in L²((0, 1)) 1.4 Decay of Fourier coefficients 1.5 The quest for convergence 1.6 Pointwise convergence of Fourier Series <"all in one breath> " 1.7 Pointwise convergence of Fourier Series: a refined version 1.8 Uniqueness results 1.9 More on the decay of Fourier coefficients 1.10 Uniform convergence results 1.11 Convergence in L²((0, 1)) 1.12 Convergence in Lp((0, 1)) 1.13 Averaging procedures for Fourier Series 1.14 Persistent overshooting phenomena 1.15 Convergence issues 1.16 Functions of arbitrary periods 1.17 Fourier Series in any dimension 2: Fourier Series: Applications 2.1 The old-fashioned epicycle theory 2.2 Does anyone need a computer? 2.3 Predicting tides 2.4 Partial differential equations 2.5 Fourier's cellar 2.6 The Dirichlet problem on the two-dimensional disk 2.7 The Dirichlet problem on a square and the behaviour near corners 2.8 Calculating the exact value of a series 2.9 Number theory 2.10 Inequalities of analytic flavour 2.11 Inequalities of geometric flavor 2.12 The Weierstrass Approximation Theorem 2.13 The Radon Transform 2.14 Filters 2.15 Edge detection 2.16 Denoising 2.17 Who listens to the radio? 2.18 Overdrives, clipping, distortion, and ringing artifacts 2.19 Linear motions on tori 2.20 Weyl's Equidistribution Theorem 2.21 Minkowski's Theorem on convex sets 2.22 The Central Limit Theorem - What Comes Next? Appendix A - One more example confirming Theorem 1.15.1 Solutions to selected exercises