Convergence of a generalized Fourier series in different basis Using a generalized Fourier series, almost every arbitrary function can be approximated as a sum using a family of orthogonal functions which I call a basis, over a domain $D$. 

For example, the traditional Fourier series utilizes a basis defined as $A = { e^{int} : n \in \mathbb{Z} }$ in the domain $[0,2\pi]$. Then the series for $f$ under $A$ is

$$S_{D} [f]_{A}(x) = \sum_{n=-\infty}^{\infty} a_{n} e^{inx}$$

However for a function such that $S_{D} [f]_{A}$ diverges almost everywhere, is it also true for all other basis or does there exist another basis $B$ with domain $H : H \cap D \neq \emptyset$ such that $S_{H} [f]_{B}$ is convergent almost everywhere in the intersection of the domains?

My personal guess is that yes there will always exist another basis such that the series for $f$ is convergent, however a proof or disproof is beyond my abilities.
Yes, there exists a basis, meaning an orthonormal basis for $L^2(D)$, such that the series for $f$ in that basis converges to $f$.

For example, take an orthonormal basis $\phi_1,\dots$ such that $\phi_1=cf$. Then the series representing $f$ is $$\sum_{n=1}^\infty c_n\phi_n=c_1\phi_1=f.$$
Yes, there exists a basis, meaning an orthonormal basis for $L^2(D)$, such that the series for $f$ in that basis converges to $f$.

For example, take an orthonormal basis $\phi_1,\dots$ such that $\phi_1=cf$. Then the series representing $f$ is $$\sum_{n=1}^\infty c_n\phi_n=c_1\phi_1=f.$$