It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint intervals $[-1,1]$ and $[a,b]$?
An upper bound can be obtained by seeking the minimal $L_\infty$ norm polynomial over the "filled-in" interval $I = [\min(-1,a), \max(1,b)]$. This would be a translated and scaled Chebyshev polynomial. My question is, can you do significantly better than this? Intuitively, can you exploit the fact that there is empty space between $[-1,1]$ and $[a,b]$ where the polynomial is not required to have a small value? Or is this empty space essentially useless?
My first instinct was to try a product of scaled Chebyshevs which are small on $[-1,1]$ and $[a,b]$ respectively. However there is no growth control on one Chebyshev in the others' interval, so there is no guarantee that you are doing any better. Because of this I am pessimistic that one can do much better than the upper bound, but I would love to be proven wrong.
I am interested in this question because I am studying the convergence of Krylov subspace methods, where such approximations play an important role. I want to understand the convergence rate of conjugate gradients when there are multiple clusters of eigenvalues contained in different intervals, rather than just a single cluster.