Imagine we want to do linear least squares fit on some interval $t\in[0,1]$, $$\min_{\bf v}\{\|{\bf \Phi v -d} \|_2^2\}$$ for a polynomial $t\to p(t)$ :
$$p(t) = \sum_{k=0}^M {\bf v}_k t^k$$
but with additional constraints:
$$p^{(k)}(0)=p^{(k)}(1) \hspace{1cm} \forall k \in \{ 0,\cdots,N \}$$
In other words no "jumps" in derivative for function or it's $N$ first derivatives.
Assuming degree of polynomial is $M$, which values of $N$ would be possible?
How restrictive would this be for our polynomial?
Given $M$ and $N$, can we find a "best basis" in some sense to represent this polynomial?
I assume in the extreme case that as $N$ goes to infinity, $M$ will necessarily need to grow to infinity as well and only degrees of freedom left would be equivalent to power series expansions for the basis function of Fourier series expansion on this interval.