Pointwise convergence of Fourier series

In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials. Euler, the Bernouillis, and others had used such sums in similar fashions and for similar ends, but Fourier made a claim extravagant for the time, namely that all functions could be expressed in such terms. Unfortunately, in those days there was no clear idea of what a function was, no vocabulary to specificy classes of functions, and no specification of what it would mean to represent a function by such a series. In hindsight, probably issues of pointwise and L convergence, unspecified to some degree, were confused with each other.