# Can "Laplace limit" phenomenon be considered the first example of non-convergent Fourier series?

Traditionally, the first example of a non-convergent Fourier series of a function is considered to be the example constructed in 1870 by the mathematician Paul David Gustav du Bois-Reymond. His example was a turning point in the study of convergence of Fourier series since it shattered the hopes expressed by previous mathematicians (including Dirichlet and Riemann) that the Fourier series of any continuous function converges into it.

However, I read in several places that in astronomy, when the difference between mean and true anomaly (in the so called "equation of the center") is developed into trigonometric series of multiples of mean anomaly with coefficients calculated by Fourier integral, the trigonometric series diverges for values of eccentricity exceeding the "Laplace limit", whose value is approximately $$0.662$$ (for more detailed explanation of the context, read my post: Meaning of passages by Gauss on the "convergence of expansions (in infinite series) of the (elliptical) equation of the center"?).

Therefore, I have much confusion about how to interpret the fact that the phenomenon of "Laplace limit" was discovered much earlier than 1870. I have never read any claims that the discovery of this phenomenon anticipated these later developments, so most probably I misunderstood the meaning of it and misevaluated the significance of discovery of Laplace limit. But the fact that I didn't find such claims is still not a conclusive proof that there is no substantial relation between the two issues.

So I'd like to put things into order and to read a correct historic assessment of the discoveries of Laplace and Paul Reymond and their relation (if there is such a relation).

• The series considered in astronomy are not exactly Fourier series; the motions they represent are not periodic, and these series usually diverge. Dec 22, 2021 at 23:54
• @Alexandre Eremenko - why the motions they represent are not periodic? If $v$ is true anomaly and $M$ is mean anomaly, than $v-M$ is periodic with the period $T$ of orbital motion. Dec 23, 2021 at 10:30
• Yes, in this case it is periodic but this is not a Fourier series. Dec 23, 2021 at 21:56
• @Alexandre Eremenko - why this is not a Fourier series? If we write $v-M$ as infinite sum of cosines of integer multiples of $M$ with coefficients $C_n$ calculated by the standard inner product between functions, than it is a Fourier series of $v-M$ with respect to variable $M$ (which itself is proportional to the time elapsed). What am I missing here? (I also believe the answer to the title question is negative, I just want to understand why). Dec 24, 2021 at 11:08