What are the $n$-th degree minimal polynomials for $L^p([-1,1])$? It is known (even by me) that the Chebyshev polynomial of degree $n$ (of the first kind) is the minimal polynomial in the space $L^{\infty}([-1,1])$ for a fixed $n$ and leading coefficient $2^{n-1}$.
However, what are the minimal polynomials for the $p$-norm in general for a fixed $n$? Does there exist a general answer?

This is my first question here and I apologize if it is not up to par. Feel free to edit, migrate or close it if necessary.
 A: It seems that your question is an open problem, but there are partial answers.
Let's fix natural number $n$, and  $1<p\leq\infty$. By $T_{n,p}$ we denote polynomial of degree $n$ with leading coefficient equa to $1$ with minimal norm in $L_p([-1,1])$.
It is known that (see On the Zeros of Polynomials of Minimal Lp-Norm
András Kroó and Franz Peherstorfer)

*

*$T_{n,1}$ is the Chebyshev polynomial of the second kind, so
$$
T_{n,1}(x)=U_n(x)=\frac{\sin((n+1)\arccos(x))}{\sin(\arccos(x))}
$$

*$T_{n,2}$ is the Legendre polynomial, so
$$
T_{n,2}(x)=L_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n
$$

*$T_{n,\infty}$ is the Chebyshev polynomial of the first kind, so
$$
T_{n,\infty}(x)=T_n(x)=\cos(n\arccos(x))
$$
However we can force Chebychev polynomials to minimize weighted norms in $L_p([0,1])$ [2]. In fact

*

*$T_n(x)$ have the smallest norm
$$
\Vert f \Vert=\left(\int\limits_{-1}^1(1-x^2)^{-1/2}|f(x)|^p dx\right)^{1/p}
$$
among polynomials of degree $n$ with leading coefficient $1$.


*$U_n(x)$ have the smallest norm
$$
\Vert f \Vert=\left(\int\limits_{-1}^1(1-x^2)^{(p-1)/2}|f(x)|^p dx\right)^{1/p}
$$
among polynomials of degree $n$ with leading coefficient $1$.
