What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max\_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$. 


Now suppose we ask the following question: what is the polynomial of degree $n$ of the form $$x^n + a\_d x^d + a\_{d-1} x^{d-1} + \cdots + a\_1 x + a\_0$$ which minimizes $\max\_{x \in [-1,1]} |p(x)|$? Here $d<n$; if we take $d = n-1$, we recover the Chebyshev polynomial. 


My main purpose in asking this is to estimate how big $\max\_{x \in [-1,1]} |p(x)|$ is as a function of $n$ and $d$ for such a polynomial. 


One can emulate the derivation of Chebyshev polynomials: we would solve the problem if we could construct a polynomial of this form which never exceeds $A$ in absolute value and oscillates between $A$ and $-A$ sufficiently many times in $[-1,1]$. However, its not immediately clear to me how to do this. 



A generalization of Chebyshev polynomials
First of all, there is a general theory (due to Chebyshev) on the best uniform approximation
of ANY continuoius function $f$ by polynomials of degree at most $d$ on an interval.
It describes the polynomial of the best approximation, which is unique.


In Chebyshev's polynomials, $f=x^n$ and $d=n-1$. You are asking about $f=x^n$ and
some given $d<n$. I do not recommend to read Chebyshev himself on this but Akhiezer, Lectures on approximation theory. Now, the case when $d=n-2$ was worked our explicitly
by Zolotarev, Chebyshev's student, in terms of elliptic functions. When $d$ is smaller than that, only some general facts about the polynomials of best approximation are known,
but I don't think anyone has written an explicit exact expression for the polynomial or for the error.


