So for a first order polynomial basis I know that the following basis is standard:
$\phi_j(t) = \begin{cases} 0 \; t<t_{i-1}, \\ \frac{t-t_{i-1}}{t_i - t_{i-1}} \; t_{i-1} \leq t \leq t_i \\ \frac{t_{i+1}-t}{t_{i+1}-t_i} \; t_i \leq t \leq t_{i+1} \end{cases}$
with a finite-element solution representation $u_h(t) = \sum_{i=1}^{N-1} c_i \phi_i(t)$. I also know that $|| u-u_h ||_{\infty} \leq \mathcal{0}(h^2)$, where is the maximum element legend e.g $h = \text{max}_i \;t_i - t_{i-1}$.
I was curious if there is a general sort of formula for how these basis functions would look if for example you wanted higher order polynomials (its cool if you need to use more elements to define those polynomails if my current support of the basis functions $[ x_{i-1},x_{i+1} ]$ is a problem). A reference would be awesome.I am particulary interested in finding a finite-element polynomial basis that would produce a $\mathcal{O}(h^4)$ error.
Also I have a conjecture that if you are representing the finite element solution with $k$ degree polynomial then $|| u-u_h ||_{\infty} \leq \mathcal{0}(h^{k+1})$. Does anyone know if this is true?
Thank you for the help.