I was thinking about the following problem the other day:
Fit a spline $s(t)$ to some transcendental function $f(t)$, so that:
$$s(t) = \cases{P_k(t), \text{ if } t_k \leq t \leq t_{k+1}}$$
For polynomials $P_k(t)$:
$$P_k(t) = \sum_{i=0}^{N} c_{ik}t^i$$
Now we seek $c_{ik}$ so that $s(t)$ approximates $f(t)$ well on interval $t\in [t_0,t_{max}]$.
How can we do this? How to choose the $t_k$ knot points?
Which boundary conditions should be obeyed there?
If the knot points were fixed then the problem could be approached like some least squares norm minimization:
$$\sum_l\|s(t_l)-f(t_l)\|_2^2 + \epsilon(\text{boundary terms})$$
All would be linear. But how to tackle the non-linearity that is introduced with not knowing where knot points should be placed?
If your question is about convergence, then interpolating splines converge to the initial function if the interpolation step converges towards zero. See for example here.