Background
I'm reading Vretblad's Fourier Analysis and its Applications. In the chapter on Fourier series, there is a section on differentiable functions.
Let $\mathbb T$ be the unit circle and denote the complex Fourier coefficient of $f$ by $c_n$. Then there is the following theorem;
Theorem 4.4 If $f\in C^k(\mathbb T)$, then $|c_n|\leq M/|n|^k$ for some constant $M$.
This theorem is not really proved in the book, but if $f$ is (Riemann) integrable over $\mathbb T$, then the Fourier coefficients are bounded. This follows from the definition of $c_n$, namely $$|c_n|=\left|\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)e^{-int}dt\right|\leq \frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f(t)\right|\left|e^{-int}\right|dt=\frac{1}{2\pi}\int_{-\pi}^{\pi} |f(t)|dt=M,$$ since $f$ is integrable over $\mathbb T$. To "prove" the theorem, if $b_n$ denotes the Fourier coefficient of $f^{(k)}$, then by recursively applying partial integration, and noting that if $g$ is continuous on $\mathbb T$, then $g(\pi)=g(-\pi)$, so \begin{align} b_n &= \frac{1}{2\pi}\int_{\mathbb T}f^{(k)}(t)e^{-int}dt \\ &= \frac{1}{2\pi}[f^{(k-1)}(t)e^{-int}]^{\pi}_{-\pi}+\frac{1}{2\pi}in\int_{\mathbb T}f^{(k-1)}(t)e^{-int} dt \\ &= \ldots \\ &=(in)^k \frac{1}{2\pi}\int_{\mathbb T}f(t)e^{-int}dt \\ &= (in)^kc_n.\end{align} Since $f^{(k)}$ is continuous (and thus integrable), we have $|b_n|\leq M$ for some $M$, i.e. $|n^k c_n|\leq M$, and the claim of the theorem follows.
Questions
There is the following exercise in the book;
Try to prove the following partial improvements of Theorem 4.4:
(a) If $f'$ is continuous and differentiable on $\mathbb T$ except possibly for a finite number of jump discontinuities, then $|c_n|\leq M/|n|$ for some constant $M$.
(b) If $f$ is continuous on $\mathbb T$ and has a second derivative everywhere except possibly for a finite number of points, where there are "corners" (i.e., the left-hand and right-hand first derivative exist but are different from each other), then $|c_n|\leq M/n^2$ for some constant $M$.
- I struggle with seeing the difference in the assumptions of these two statements. Is (a) not assuming the same as (b)?
- Consider statement (a) and the assumptions on $f'$. What does this tell us about $f$? I've been trying to compute the Fourier coefficients of $f'$ as above for the "proof" of theorem 4.4, i.e. via partial integration, but I'm not sure what properties $f$ has.
- Any hints for (b)?
Grateful for any help.