We consider the problem of global uniform convergence of spectral expansions and their derivatives, $\sum\limits_{n=1}^{\infty}f_n\,u^{(j)}_n(x)$ ($j=0,1,\dots$), generated by arbitrary self-adjoint extensions of the operator $\mathcal L(u)(x) = - u''(x) + q(x)\,u(x)$ with discrete spectrum, for functions from the classes $H_p^{(k,\alpha)}(G)$ ($k\in \mathbb N$, $\alpha\in (0,1]$) and $W^{(k)}_p(G)$ ($1\le p\le 2$), where $G$ is a finite interval of the real axis. Two theorems giving conditions on functions $q(x)$, $f(x)$ which are sufficient for the absolute and uniform convergence on $\olG$ of the mentioned series, are proved. Also, some convergence rate estimates are obtained.