Question
Does there exist general closed form solutions (or some sort of recurrence relation) to the system of equations: $$\begin{align} x_0 &= -1\\ x_{k+1} &= 1\\ \sum_{j = 0}^k (-1)^j (x_{j+1} - x_{j}) & = 0 \\ \sum_{j = 0}^k (-1)^j (x_{j+1}^2 - x_{j}^2) &= 0 \\ \vdots \\ \sum_{j = 0}^k (-1)^j (x_{j+1}^k - x_{j}^k) &= 0 \end{align}$$ with $-1 \leq x_1 \leq x_2 \leq \ldots \leq x_k \leq 1$?
For $k$ small the solutions can be computed by hand to be
- $k = 1$: $\{0\}$
- $k = 2$: $\{\pm \frac12\}$
- $k = 3$: $\{0, \pm\frac12\}$
- $k = 4$: $\{(\pm 1\pm\sqrt{5})/4\}$
- $k = 5$: $\{0, \pm\frac12, \pm\frac{\sqrt{3}}2\}$
The non-trivial part of the system can be summarised as $$ \begin{pmatrix} x_0 & x_1 & x_2 & \ldots & x_k & x_{k+1} \\ x_0^2 & x_1^2 & x_2^2 & \ldots & x_k^2 & x_{k+1}^2 \\ x_0^3 & x_1^3 & x_2^3 & \ldots & x_k^3 & x_{k+1}^3 \\ & \vdots \\ x_0^k & x_1^k & x_2^k & \ldots & x_k^k & x_{k+1}^k \end{pmatrix} \begin{pmatrix} -1 \\ 2\\ -2 \\ 2 \\ -2 \\ \vdots \\ (-1)^{k-1} 2\\ (-1)^k\end{pmatrix} = 0 $$ if it helps.
Motivation
This question asked for the minimizer among linear functions of the functional $\int_0^1 |e^x - \ell(x)|dx$. The solution given by anonymous user surprised me slightly: it turns out that there is a universal construction that is independent of the target function (in this case, $e^x$). Namely, the user has actually proven that
Theorem Let $f\in C^2([-1,1])$ such that $f''$ is strictly signed. Then the minimizer of the functional $\int_{-1}^1 |f(x) - \ell(x)|dx$ among linear functions $\ell(x)$ is attained when $$ \ell(x) = (x + 1/2) \left( f(1/2) - f(-1/2)\right) + f(-1/2) $$ in other words, it is the linear interpolation function between $(-1/2, f(-1/2))$ and $(1/2, f(1/2))$.
I gave another write-up of the proof here on my blog.
It turns out that the theorem extends to the case of "higher order convexity", in the following sense:
Theorem There exists a list of numbers $\{x_1, \ldots, x_k\}\subset [-1,1]$ such that for every $f\in C^k([-1,1])$ such that $f^{(k)}$ is strictly signed, the minimizer of the functional $\int_{-1}^1 |f(x) - p(x)|dx$ among polynomials of degree at most $k-1$ is attained by the $(k-1)$-order polynomial interpolation based on the points $(x_j, f(x_j))$.
I wrote up a proof again on my blog.
The question above is asking for general closed form expressions of the numbers $\{x_1, \ldots, x_k\}$ in terms of $k$.
(If someone happens to know a literature reference for this result, that would also be very nice.)
First off, your answer for $k=3$ is wrong -- the correct values are $\{0, \pm \frac{1}{\sqrt{2}} \}$.
Second, the answer is $x_{k+1-m} = \cos{\frac{m\pi}{k+1}}, \, m=0\ldots k+1$.