Why does the B-spline constructed from squaring its coefficients appear to converge to the square of the B-spline itself?
To be clear, I am not sure if this occurs in all situations, but it appears to occur for some simple computational tests that I've run. In these tests, I approximate a given function by a one dimensional B-spline. The B-spline's knot vector is not necessarily uniform, but the continuity is chosen to be fixed across the whole domain. Meaning, there is a fixed and specified continuity across the mesh, so the whole spline is in something like $C^2$ or $C^3$. By mesh, I mean the knot vector without repeated knots. The approximation is generated by matching the function at the Gaussian quadrature points mapped to the mesh. Refinement occurs by refining the mesh/knot vector.
In any case, very quickly the B-spline converges to the function at machine precision. More slowly, taking the coefficients, squaring them, and then constructing a new B-spline based on these squared coefficients appears to converge to the square of the function in the $L^2$-norm. I'm interested to hear under which conditions this is expected to occur and, if possible, how quickly it should be expected to converge.
Edit 1
Per request, here is an example of the behavior. The following is an approximation of the function $t\sqrt{\cos(2\pi t)+1.5}-3$ using an order 20 B-spline on a uniform mesh where the knots are chosen to maintain a function in $C^5$, or 5 continuous derivatives between the mesh points. Basically, the knots are repeated enough times to ensure a continuously differentiable function. The function itself can be seen here:
The $L^2$ error in the approximation is seen here:
Recall, the approximation is made by matching the spline to the function at collocation points (Gaussian quadrature points) mapped to the mesh. The final graph is the $L^2$ error between the square of the B-spline approximation and a B-spline formed by squaring the coefficients:
A few things to note, it appears to be converging, but slowly. Also, the convergence occurs even after the original approximation hits essentially machine precision in its approximation of the original function.
Thanks for any insight.