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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 $2^n$.

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.

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    @Norbert So basically it's an open question besides the two kinds of Chebyshev polynomials and Legendre polynomials (I only read the first page, I don't have an account in JSTOR)? If so, I'd accept an answer from you.2012-06-08

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It seems that your question is an open problem, but there are partial answers.

Let's fix natural number $n$, and $1. 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 [1]

  • $T_{n,1}$ 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 $1$.