Consider the following set of Bessel functions
$$\{J_1(\alpha_ir)\}, \qquad J_0(\alpha_ia)=0 \tag{1}$$
I want to show that this set of functions form a basis for the space of $C^{1}[0,a]$ functions. So I should prove that
They are linearly independent and that they span the space so they form a basis for that space.
My Work
My first thought was to find the corresponding Sturm-Liouville problem for this set of functions. However, I failed to find proper boundary conditions. For example, the following Sturm-Liouville system
\begin{align}
\,&\frac{d}{dr}\left[r\frac{dR}{dr}\right]+\left[\lambda r + \frac{1}{r}\right]R=0\\
&R(0)<\infty\\
&R(a)=0
\tag{2}
\end{align}
leads to the following basis
$$\{J_1(\alpha_ir)\}, \qquad J_1(\alpha_ia)=0 \tag{3}$$
for the aforementioned space (note the order of the Bessel functions).
Anyway, I could show that the set of functions mentioned in Eq.$(1)$ are orthogonal and consequently linearly independent by just computing the following integral
$$\int_{0}^{a} r J_1(\alpha r) J_1(\beta r)dr = \frac{a}{\alpha^2-\beta^2}\Big(\beta J_0(\beta a)J_1(\alpha a)- \alpha J_0(\alpha a)J_1(\beta a)\Big)$$
Best Answer
I verified that the Fourier Bessel on $(0,b)$ converges in the same way that the ordinary Fourier series on the same interval does. Titchmarsh was the old master of pointwise convergence proofs for general eigenfunction expansions of second order ODEs. I recommend always starting with his work when looking at the subject of pointwise convergence for generalized Fourier integral and series expansions.
Reference: E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, Second Edition, 1962, Chapter 4, page 84.