ABSTRACT
Throughout the eighteenth century, leading mathematicians became aware that power series were insufficient to represent functions, and that a different type of series was needed. Considerations of physical problems, such as behavior of a vibrating string, suggested that their terms should be trigonometric functions. Much of the nineteenth century mathematics has its roots in the problems associated to Fourier series. Mathematicians were thus left with the challenge to explain the success of the Fourier’s theory. This had a profound impact on the next 100 years of mathematical research, stretching all the way to the present. Clearly, the definition of a function had to be freed from its algebraic shackles, and the modern definition of the function emerged through the work of Dirichlet and Riemann. Pointwise convergence was not the only hot topic in the study of Fourier series. When such a series converges it is of interest to determine the properties of the limit function such as continuity or differentiability.
