When iteratively findind the best polynomial approximation wrt. $\left\|\cdot\right\|_{\infty}$, a good starting point for the process is the projection onto the basis of Chebyshev 1st kind polynomials, because their monic multiple have minimal $L_{\infty}$ norm. Similarly a Legendre projection is used for an $L_2$ approximation.
However, I think it maybe is better to use the projection onto the family of polynomials that has minimal $L_p$ norm for the multiple with $P_n(1)=1$ rather than the monic multiple. For $p=\infty$ it makes no difference, but for $p=2$ it does.
For $p=2$, the family has this 3-term recurrence relation:
$$P_0(x) = 1\quad,\quad P_1(x) = x\quad,\quad P_{n+1}(x) = \frac{3 + 2n}{3 + n}xP_{n}(x) - \frac{n}{3 + n} P_{n-1}(x)$$
The roots of $P_n(x)$ are the eigenvalues of $M_n$, where $$M_n=\left( \begin{array}{cccccc} 0 & b(1) & 0 & 0 & 0 & 0 \\ b(1) & 0 & b(2) & 0 & 0 & 0 \\ 0 & b(2) & 0 & \ddots & 0 & 0 \\ 0 & 0 & \ddots & 0 & \ddots & 0 \\ 0 & 0 & 0 & \ddots & 0 & b(n-1) \\ 0 & 0 & 0 & 0 & b(n-1) & 0 \\ \end{array} \right)\quad,\quad b(n)=\sqrt{n/(3/(n+2)+4n)}$$ Is this a named/wellknown family? If no, how is the weight function related to the recurrence relation and $M_n$?
For $p = 2$ the recurrence relation has Gegenbauer polynomials as solutions. First normalize the recurrence relation to $$ x p_n(x) = p_{n+1}(x) + \frac{n(n+2)}{(2n+1)(2n+3)} p_{n-1}(x). $$ Compare with the normalized recurrence relation for the Jacobi polynomials, so that $$ p_n(x) = \frac{2^n n!}{(n+3)_n}P^{(1,1)}_n(x). $$ The Jacobi polynomials for which $\alpha = \beta$ can be written as Gegenbauer polynomials. Therefore with $\lambda = \frac{3}{2}$, $$ P^{(1,1)}_n(x) = \frac{(2)_n}{(3)_n} C_n^{(\frac{3}{2})}(x) = \frac{2}{n+2} C^{(\frac{3}{2})}_n(x). $$ The weight function for these Gegenbauer polynomials is $w(x) = 1 - x^2$.
Best, Noud