Let $\{e_k\}$ be a polynomial orthonormal complete basis of $L^2(0,1)$. Let $f:(0,1) \rightarrow \mathbb R$. The generalized Fourier series of $f$ for the basis $\{e_k\}$, when it exists, is denoted by
$$\forall x \in (0,1), \quad S(x)=\lim_{n\rightarrow \infty} S_n(x)$$
$$S_n(x)= \sum_{k=1}^n\langle f, e_k\rangle_{L^2} e_k(x)$$
1) Suppose $f\in L^2(0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\forall x, \lim_{n\rightarrow \infty} S_n(x) =f(x)$ ?
2) Suppose $f\in L^2(0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\lim_{n\rightarrow \infty} S_n(x)= f(x)$ if $x$ is a continuity point of $f$ and $\lim_{n\rightarrow \infty} S_n(x)= \frac{f(x^+)+f(x^-)}{2}$ otherwise ?
3) Suppose $f\in L^2(0,1)$, increasing and left-continuous with right limits everywhere on $(0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\lim_{n\rightarrow \infty} S_n(x)= f(x)$ if $x$ is a continuity point of $f$ and $\lim_{n\rightarrow \infty} S_n(x)= \frac{f(x^+)+f(x^-)}{2}$ otherwise ?
4) Suppose $f\in L^1(0,1)$ and $S(x)$ exists $\forall x \in (0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\forall x, \lim_{n\rightarrow \infty} S_n(x) =f(x)$ ?
5) Suppose $f\in L^1(0,1)$ and $S(x)$ exists $\forall x \in (0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\lim_{n\rightarrow \infty} S_n(x)= f(x)$ if $x$ is a continuity point of $f$ and $\lim_{n\rightarrow \infty} S_n(x)= \frac{f(x^+)+f(x^-)}{2}$ otherwise ?
6) Suppose $f\in L^1(0,1)$, increasing and left-continuous with right limits everywhere on $(0,1)$, and $S(x)$ exists $\forall x \in (0,1)$. What extra conditions (as minimal as possible) should we put on $f$ to get $\lim_{n\rightarrow \infty} S_n(x)= f(x)$ if $x$ is a continuity point of $f$ and $\lim_{n\rightarrow \infty} S_n(x)= \frac{f(x^+)+f(x^-)}{2}$ otherwise ?
Any reference that includes proofs about different types of convergences of generalized Fourier series is very welcome.