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6. Ideas that do not seem to work

It may be good to have a list of tries that do not seem to succeed.
    1. Problem 6.1.

      [N. Trefethen] Given $f$ with $\|f\|_{\infty,W(A)}=1$, can we always find $\tilde{f}$ defined on a ball $D$ enclosing $W(A)$ such that $\tilde{f}\approx f$ on $W(A)$ and $\|\tilde{f}\|_{\infty,D}=1$?
          Ideally, $\tilde{f}=f\phi$ for some function $phi$ that is approximately equal to $1$ on $W(A)$ and well-behaving outside. If the above was true, then the conjecture should be clear.

          Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.