A MCQ question about Fourier series appeared in NBHM PhD entrance $2018.$ Source: National Board for Higher Mathematics Model Questions, PhD Jan 2018 entrance exam paper, Q2.7  (Direct PDF  link, answer key link)

Let $f : \left [-\pi, \pi \right ] \longrightarrow \Bbb R$ be a continuous $2\pi$-periodic function whose Fourier series is given by $$\dfrac {a_0} {2} + \sum\limits_{k=1}^{\infty} \left (a_k \cos kt + b_k \sin kt \right ).$$ Let, for each $n \in \Bbb N,$ $$f_n (t) = \dfrac {a_0} {2} + \sum\limits_{k=1}^{n} \left (a_k \cos kt + b_k \sin kt \right ),$$ and $f_0$ denote the constant function $\dfrac {a_0} {2}.$ Which of the following statements are true?
a. $f_n \to f$ uniformly on $\left [-\pi, \pi \right ].$
b. If $\sigma_n = \dfrac {f_0 + f_1 + \cdots + f_n} {n + 1},$ then $\sigma_n \to f$ uniformly on $\left [-\pi, \pi \right ].$
c. $\displaystyle {\int_{-\pi}^{\pi} {\left \lvert f_n (x) - f(x) \right \rvert}^2\ dx \to 0,}$ as $n \to \infty.$

If $f'$ is piecewise smooth $2\pi$-periodic function then option a
is true. What will happen if $f'$
is not given to be piecewise smooth? Is there any counter-example? Also I don't know anything about the last two options.
How do I proceed? Any help will be highly appreciated.
Thanks in advance.
 A: Both b and c are correct.
Option a is not correct. There are continuous functions whose fourier series do not converge pointwise. Regarding your comment on $f'$ being piecewise smooth being enough, actually you only need a small amount of Hölder regularity say $f\in C^\alpha$ for some $\alpha\in(0,1)$ to get uniform convergence.
For b, this follows from the fact that Fejer kernels are "good kernels", and so you have an "approximation to the identity" (these are Googling keywords). One place to find the proof of b is here (random Google result.)
For c, this follows because continuous functions on $[-\pi,\pi]$ (periodic or not) are in particular $L^2([-\pi,\pi])$ functions, and here, the exponential functions are a complete orthonormal set / Schauder basis. Thus the partial expansion converges in $L^2$ norm, and to the original function $f$, which is exactly the statement of c. One can look here for a proof. (There is the related and harder result called Carleson's theorem which says that Fourier series of $L^2$ functions even converges almost everywhere.)
If you prefer a book I would imagine these are covered (at a sufficiently 'elementary' level) in one of Stein and Shakarchi's books.
A: I think choice c is correct which is basically the whole motivation behind the study of Fourier series. They approximate a function using sinusoids.
