# Dirichlet's Proof of the Convergence of Fourier Series

Where can I find Dirichlet's proof of the convergence of Fourier series?

• Are you asking for the original source or just some exposition of what textbooks call "Dirichlet's theorem"? – Conifold Aug 22 '18 at 21:55
• @Conifold Both. – user109871 Aug 23 '18 at 12:22
• Possible duplicate of Dirichlet integral's history – Big Brother Jul 14 '19 at 2:19
• What do you have against Harmonic Analysis? – Rodrigo de Azevedo yesterday

## 2 Answers

Dirichlet's proof of the convergence of the Fourier series was nearly identical to a proof that Fourier offered in his original treatise on Heat Conduction. The reason this seems to have gone unnoticed is that Fourier's original manuscript was banned from publication for over a decade. Dirichlet was Fourier's student, and probably had access to the unpublished manuscript. H.S. Carslaw notes this in the Historical Introduction to his 1906 book Introduction to the Theory of Fourier’s series and integrals:

"However, it is a mistake to suppose that Fourier did not establish in a rigorous and conclusive manner that a quite arbitrary function (meaning by this any function capable of being represented by an arc of a continuous curve or by successive portions of diﬀerent continuous curves), could be represented by the series with his name, and it is equally wrong to attribute the ﬁrst rigorous demonstration of this theorem to Dirichlet, whose earliest memoir was published in 1829. A closer examination of Fourier’s work will show that the importance of his investigations merits the fullest recognition, and Darboux, in the latest complete edition of Fourier’s mathematical works points out that the method he employed in the ﬁnal discussion of the general case is perfectly sound and practically identical with that used later by Dirichlet in his classical memoir. In this discussion Fourier followed the line of argument which is now customary in dealing with inﬁnite series. He proved that when the values $$a_n,b_n$$ are inserted in the terms of the series" $$a0 + (a1 \cos x + b1 \sin x) + (a2 \cos 2x + b2 \sin 2x) + …,$$ the sum of the terms up to $$\cos(nx)$$ and $$\sin(nx)$$ is $$\frac{1}{\pi}\int_{-\pi}^{\pi} f(x')\frac{ \sin \frac{1}{2}(2n + 1)(x'−x) }{\sin \frac{1}{2}(x'−x)} dx'$$ He then discussed the limiting value of this sum as n becomes inﬁnite, and thus obtained the sum of the series now called Fourier’s Series."

• Interesting, do you know a page range for the original proof of Fourier in his treatise? – Michael Bächtold Nov 2 '18 at 9:24

The first line and footnote of this answer link Dirichlet’s paper (in French) as well as a translation and expositions in English and German.

• I am thinking we should close both these questions as duplicates. – Conifold Aug 25 '18 at 3:50