Abstract. Letϕmbeapolynomial satisfyingsomemildconditions. Givenaset RC, acontinuousfunction 

fonRanditsbestapproximationp 

n−1fromΠn−1withrespect tothemaximumnorm,weshowthat pn−1◦ϕm 

isabest approximationtof◦ϕmontheinversepolynomial imageSofR, i.e. ϕm(S)=R, wheretheextremal 

signatureisgivenexplicitly. Asimilar result ispresentedfor constrainedChebyshevpolynomial approximation. 

Finally, weapplytheobtainedresults tothecomputationof theconvergence rateof Krylovsubspace methods 

whenappliedtoapreconditionedlinear system. Weinvestigatepairsofpreconditionerswheretheeigenvaluesare 

containedinsetsSandR, respectively, whicharerelatedbyϕ(S)=R.