A rigorous introduction to the convergence of Fourier series

Question

I have been trying to find a resource on Fourier series and their convergence that is rigorous, does not leave major steps to the reader, and does not require a background in Lebesgue integration. Ideally this would be a set of computer verified formal proofs presented with accompanying text that could be read at an undergraduate level. Is there any chance that something like this might exist?

Answer 1

To my knowledge and experience, the best exposition I ever found about the convergence of the Fourier series is that provided by Kolmogorov & Fomin in the relevant chapter of their Elements of the Theory of Functions and Functional Analysis , among all the rest of the book that is definitely worthy to study carefully.

Their analysis starts from the Dirichlet Kernel and relevant integral, and goes through the Fejer's and Weierstrass theorems.

Answer 2

An old classic that many people used is the Fourth Edition of A Course of Modern Analysis by E.T. Whittaker and G.N. Watson. Chapter IX is Fourier Series and Trigonometrical Series. They use only classical methods of analysis to study convergence