Let $\{\phi_n\}_{n=0}^\infty$ be the eigenfunctions of the regular Sturm-Liouville problem \begin{align} -(p\,\phi')' + q\, \phi = \lambda \, r \, \phi \quad &\textrm{for } x \in (x_1,x_2)\\ - a_i \, \phi(x_i) + b_i\, (p\,\phi')(x_i) = 0 \quad &\textrm{for } i=1,2. \end{align} Assume that $p$ and $r$ are positive and twice continuously differentiable; assume that $q$ is continuous; the coefficients $a_i,b_i$ for $i=1,2$ are real.
Let $F(x)$ be a twice continuously differentiable function on the interval $[x_1,x_2]$. Under the above conditions, I know that \begin{equation} \textrm{(I)} \quad \quad F(x) = \sum_{n=0}^\infty \left(\int_{x_1}^{x_2} F(z) \, \phi_n(z)\, r(z)\, \textrm{dz} \right)\, \phi_n(x) \end{equation} with point-wise equality in the open interval $(x_1,x_2)$.
My question is: to what value does the end-point $(x=x_i)$ series \begin{equation} \textrm{(II)} \quad \quad \sum_{n=0}^\infty \left(\int_{x_1}^{x_2} F(z) \, \phi_n(z)\, r(z)\, \textrm{dz} \right)\, \phi_n(x_i) \end{equation} converge to? Is there a general closed form expression?
If $F(x)$ satisfies the same boundary conditions as the eigenfunctions $\phi_n$, then I know that the series (I) converges to $F(x)$ uniformly on the closed interval $[x_1,x_2]$ (and so I obtain point-wise equality on the closed interval).
On the other hand, if the eigenfunctions $\phi_n$ satisfy the simpler boundary conditions $\phi_n(x_i)=0$ then the endpoint series (II) must converge to zero. The series (I) must then have a finite-jump discontinuity at the end points, e.g., jumping from $\lim_{x\rightarrow x_2}F(x)$ to $0$ at $x=x_2$. However, I am interested in the more general boundary conditions above.
I am aware of closed-form expressions for the endpoint series in the case of a Fourier expansions; I am wondering whether an analogous expression exists for regular Sturm-Liouville expansions.
Any references would be greatly appreciated.
Edit: I've linked a related question here. Is there an analogous result for Sturm-Liouville series? Do we obtain point-wise convergence to $F(x)$ on the closed interval $[x_1,x_2]$ whenever $b_1,b_2 \neq 0$?
Edit #2: The Sturm-Liouville article on the Encyclopedia of Mathematics states that, with $b_1,b_2 \neq 0$, the expansion (I) converges under the same conditions as a Cosine series for any $F\in L^1$. Presumably, from the previous edit, this would imply that we obtain point-wise convergence to $F$ on the whole interval if $F$ is differentiable and $b_1,b_2 \neq 0$. Unfortunately, I do not have access to the articles cited in the encyclopedia.
In section 9 of the Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators by Levitan and Sargsjan, the authors show that for the Sturm-Liouville problem \begin{align} -y'' + q\,y = \lambda \,y\\ y'(0) - h\, y(0) = 0 \\ y'(\pi) + H\, y(\pi) = 0 \end{align} on $[0,\pi]$, if $h,H \neq \infty$, then the Sturm-Liouville eigenfunction expansion converges or diverges at any point in the closed interval $[0,\pi]$ according to the behaviour of the corresponding cosine series expansion.
If one of $h$ or $H$ is infinity, one must instead compare with the $sin([n+1/2]x)$ expansion. Otherwise, if both $h=\infty, H=\infty$, then one must compare with the sine series expansion.