## Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods

Bernd Fischer and Franz Peherstorfer

### Abstract

Let $\varphi_m$ be a polynomial satisfying some mild conditions. Given a set $R \subset {\bf C}$, a continuous function $f$ on $R$ and its best approximation $p_{n-1}^*$ from $\Pi_{n-1}$ with respect to the maximum norm, we show that $p_{n-1}^* \circ \varphi_m$ is a best approximation to $f \circ \varphi_m$ on the inverse polynomial image $S$ of $R$, i.e. $\varphi_m(S)=R$, where the extremal signature is given explicitly. A similar result is presented for constrained Chebyshev polynomial approximation. Finally, we apply the obtained results to the computation of the convergence rate of Krylov subspace methods when applied to a preconditioned linear system. We investigate pairs of preconditioners where the eigenvalues are contained in sets $S$ and $R$, respectively, which are related by $\varphi_m(S)=R$.

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### Key words

Chebyshev polynomial, optimal polynomial, extremal signature, Krylov subspace method, convergence rate.

### AMS subject classifications

41A10, 30E10, 65F10.