This paper investigates the optimal Hermite interpolation of Sobolev spaces W∞n[a,b]\documentclass[12pt]{minimal}
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\begin{document}$W_{\infty }^{n}[a,b]$\end{document}, n∈N\documentclass[12pt]{minimal}
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\begin{document}$n\in \mathbb{N}$\end{document} in space L∞[a,b]\documentclass[12pt]{minimal}
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\begin{document}$L_{\infty }[a,b]$\end{document} and weighted spaces Lp,ω[a,b]\documentclass[12pt]{minimal}
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\begin{document}$L_{p,\omega }[a,b]$\end{document}, 1≤p<∞\documentclass[12pt]{minimal}
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\begin{document}$1\le p< \infty $\end{document} with ω a continuous-integrable weight function in (a,b)\documentclass[12pt]{minimal}
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\begin{document}$(a,b)$\end{document} when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L∞\documentclass[12pt]{minimal}
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\begin{document}$L_{\infty }$\end{document} (or Lp,ω[a,b]\documentclass[12pt]{minimal}
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\begin{document}$L_{p,\omega }[a,b]$\end{document}, 1≤p<∞\documentclass[12pt]{minimal}
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\begin{document}$1\le p<\infty $\end{document}) are optimal for W∞n[a,b]\documentclass[12pt]{minimal}
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\begin{document}$W_{\infty }^{n}[a,b]$\end{document} in L∞[a,b]\documentclass[12pt]{minimal}
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\begin{document}$L_{\infty }[a,b]$\end{document} (or Lp,ω[a,b]\documentclass[12pt]{minimal}
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\begin{document}$L_{p,\omega }[a,b]$\end{document}, 1≤p<∞\documentclass[12pt]{minimal}
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\begin{document}$1\le p<\infty $\end{document}). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.
