AbstractThis paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$
W
∞
n
[
a
,
b
]
, $n\in \mathbb{N}$
n
∈
N
in space $L_{\infty }[a,b]$
L
∞
[
a
,
b
]
and weighted spaces $L_{p,\omega }[a,b]$
L
p
,
ω
[
a
,
b
]
, $1\le p< \infty $
1
≤
p
<
∞
with ω a continuous-integrable weight function in $(a,b)$
(
a
,
b
)
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 $L_{\infty }$
L
∞
(or $L_{p,\omega }[a,b]$
L
p
,
ω
[
a
,
b
]
, $1\le p<\infty $
1
≤
p
<
∞
) are optimal for $W_{\infty }^{n}[a,b]$
W
∞
n
[
a
,
b
]
in $L_{\infty }[a,b]$
L
∞
[
a
,
b
]
(or $L_{p,\omega }[a,b]$
L
p
,
ω
[
a
,
b
]
, $1\le p<\infty $
1
≤
p
<
∞
). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.