The purpose of this paper is to explore the basic question of the convergence of Fourier series. This paper will not delve into the deeper questions of convergence that measure theory illuminates, but requires only the basic principles set out by introductory real and complex analysis.

Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propagation in bodies, Fourier was led to claim that “every” function could be represented by an infinite series of elementary trigonometric functions — sines and cosines. A...