Continuity/smoothness in terms of Fourier series

Stream: Is there code for X?

[Yury G. Kudryashov]

Do we have theorems saying that for a periodic function,

• if the Fourier series converges sufficiently slowly, then the function can't be continuous;
• if the Fourier series converges sufficiently fast, then the function is infinitely smooth?

[Antoine Chambert-Loir]

Do you mean, in math or in Lean? I fear that there are no theorems of the first kind because of a theorem by de Leeuw, Katznelson and Kahane. https://zbmath.org/0372.42004

[Antoine Chambert-Loir]

https://gallica.bnf.fr/ark:/12148/bpt6k5786410h/f53.item

[Yury G. Kudryashov]

For the first question, I also know that the coefficients are positive.

[Yury G. Kudryashov]

So, the question is: do we have the theorem saying that the Fourier series of a continuous function is Cesàro summable?