Pointwise convergence in a Sturm-Liouvlle problem.

Question

Let's consider a Sturm-Louiville problem in $[0,1]$. For me it's clear that a Fourier series of eigenfunctions converges uniformely for any continously differentiable function $F$ with $F(0)=F(1)=0$ and second derivative piecewise continous. Equivalentily, for the Green's Function $G(s,t)$, there is a piecewise continous $f(t)$ such that $ F(s) = \int_0^1 G(s,t) f(t) dt$. Said that, I don't know in which conditions pointwise convergence can be garanteed. I can give for example $G(s,t)$ for a given $s$. It is continously differentiable execpt at $t=s$, where there is a jump discontinuity in the derivative. How can I prove pointwise convergence in a case like that? You can assume that the eigenvalues are all positive.

Answer 1

Here's a Theorem from the first chapter of E. C. Titchmarsh's book on Eigenfunction Expansions:

1.9 Theorem Let $f(y)$ be integrable over $(a,b)$. Then if $a < x < b$ the Sturm-Liouville expansion behaves as regards convergence in the same way as an ordinary Fourier series. In particular, it converges to $\frac{1}{2}\{f(x+0)+f(x-0)\}$ if $f(x)$ if of bounded variation in the neighbourhood of $x$.