Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him to set theory. I'd like to know a bit more about it, yet trying to look up impact of Fourier series inevitably brings me to signal processing and alike. Where could I read an account of Fourier series' impact within mathematics?
I've found a few references, e.g. the section titled "The impact of Fourier series on mathematical analysis" in From the Calculus to Set Theory, 1630-1910: An Introductory History (edited by I. Grattan-Guinness) but it wasn't what I expected.
I think Lebesgue's Theory of Integration: Its Origins and Development by Thomas Hawkins might be what you're looking for.