Let $f,g\in R(\mathbb T)$, and $\hat{f}(n)^{2/3}=\hat{g}(n)$. How can I show absolute and unifrom converges of the Fourier series expansion of $f: S_Nf$?
From the equality above, and since both $\hat{f}(n),\hat{g}(n) \to 0$ from Riemann–Lebesgue lemma, I can get that (by switching side) $\hat{g}(n)=o(n^{2/3})$. But unfortunately since $\frac{2}{3}<1$ I couldn't use that to show that the sum converges. I also tried to show that $\left|\sum\hat{f}(n)\right|<\infty$ but couldn't manage to do that also. Would love help on this one, thanks!