Fourier Series

Joseph Fourier is one of the most famous mathematicians. For an introduction to his life, his influence, and the late recognition of his importance, we refer to the beautiful texts of Jean-Pierre Kahane [90, 91, 93]. He is mainly known for having introduced the decomposition of a periodic function into an infinite sum of trigonometric functions of frequencies each of which is a multiple of a fundamental frequency. He had conjectured that every function was representable by a convergent trigonometric series, which is incorrect.We will see counterexamples to the pointwise convergence of Fourier series and will show results on the convergence of Fourier series in the context of integrable functions.