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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.

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  • $\begingroup$ Take a look at the Historical Introduction Chapter of H.S. Carslaw's Introduction to the Theory of Fourier's Series and Integrals. He details the controversy over expanding an arbitrary function in a trigonometric series, and how Lagrange and others were strongly opposed to Fourier's conjectures. Trying to address Fourier's assertions was a key force behind the development of rigorous Math. There's also information in History of Functional Analysis by J. Dieudonne about how Fourier's work led to Spectral Theory. $\endgroup$
    – Disintegrating By Parts
    Oct 23, 2014 at 19:11
  • $\begingroup$ Trigonometric Series and the Beginnings of Set Theory. The linked paper is here. $\endgroup$
    – Ooker
    Jun 14, 2018 at 16:07

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I think Lebesgue's Theory of Integration: Its Origins and Development by Thomas Hawkins might be what you're looking for.

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