General reparameterization of a B-spline Say I have a B-spline function (or curve) of order $k_1$, defined over some knot vector
$\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e.  $$f(x) = \sum_i a^i B_{i,k_1}(x).$$
Do you know of a process of finding another B-spline function, say $g(u) = \sum_j b^j B_{j,k_2}(u)$, of order $k_2$ defined on some other knot vector $\mathbf{\xi} = \{ \xi_i\}_1^{n_2}$ that results from some arbitrary reparameterization $x \equiv x(u)$ of $f(x)$? That is:  $$g(u) = f(x(u)) \longrightarrow \sum_j b^j B_{j,k_2}(u) = \sum_i a^i B_{i,k_1}(x(u))$$
I know there are well-defined methods for finding the B-spline representation of $g(u)$ (i.e. exact evaluation of coefficients $b^i$ and B-spline basis $B_{i,k_2}(u)$) if the reparametrrization $x(u)$ is a polynomial, or another B-spline function, e.g. $x(u) = \sum_i c^i B_{i,k_3}(u)$. I am interested in a general method that can handle an arbitrary reparameterization, e.g. something like
$$ x(u) = \sqrt{u^2 + 1}.$$ I am interested for an analytic method/evaluation of $b^j$ and $\mathbf{\xi} = \{ \xi_i\}_1^{n_2}$ from knowledge of $\mathbf{t} = \{ t_i\}_1^{n_1}$ and $a^i$, not for an approximate numerical solution. Thank you very much for your time.
 A: B-splines are a basis-function representation for piecwise polynomial functions. Therefore, if the reparameterization you seek cannot be represented as a piecewise polynomial it cannot, in general, be represented as a B-spline.
This can be shown with the example you gave. For the reparameterization $x(u) = \sqrt{u^2+1}$, even a simple spline function such as $f(x) = x$ cannot be represented by a polynomial.
The reparameterization will give us:
$f(u) = f(x(u)) = x(u) = \sqrt{u^2+1}$
Since, the square root function cannot, in general, be represented as a polynomial, we will not be able to represent this function as a B-spline function no matter what the parameterization will be.
As noted in the question, if the reparameterization is polynomial the composition of the function with the reparameterization will be polynomial and can be represented as a B-spline. If the reparameterization is a spline, then the composition can be divided (e.g., using knot insertion) into intervals in which the function is polynomial, and therefore can also be represented as a B-spline. 
So, for special cases where the reparameterization can be represented as a (piecewise) polynomial function, the reparameterization is possible, but for the general case it is not.  
