Let
w
λ
(
x
)
:=
(
1
−
x
2
)
λ
−
1
/
2
w_{\lambda }(x):=(1-x^2)^{\lambda -1/2}
and
P
n
(
λ
)
P_n^{(\lambda )}
be the ultraspherical polynomials with respect to
w
λ
(
x
)
w_{\lambda }(x)
. Then we denote by
E
n
+
1
(
λ
)
E_{n+1}^{(\lambda )}
the Stieltjes polynomials with respect to
w
λ
(
x
)
w_{\lambda }(x)
satisfying
∫
−
1
1
w
λ
(
x
)
P
n
(
λ
)
(
x
)
E
n
+
1
(
λ
)
(
x
)
x
m
d
x
{
=
0
,
a
m
p
;
0
≤
m
>
n
+
1
,
≠
0
,
a
m
p
;
m
=
n
+
1.
\begin{eqnarray*} \int _{-1}^1 w_{\lambda }(x) P_n^{(\lambda )}(x)E_{n+1}^{(\lambda )}(x) x^m dx \begin {cases} =0, & 0 \le m > n+1,\\ \neq 0, & m=n+1. \end{cases} \end{eqnarray*}
In this paper, we show uniform convergence of the Hermite–Fejér interpolation polynomials
H
n
+
1
[
⋅
]
H_{n+1}[\cdot ]
and
H
2
n
+
1
[
⋅
]
{\mathcal H}_{2n+1}[\cdot ]
based on the zeros of the Stieltjes polynomials
E
n
+
1
(
λ
)
E_{n+1}^{(\lambda )}
and the product
E
n
+
1
(
λ
)
P
n
(
λ
)
E_{n+1}^{(\lambda )}P_n^{(\lambda )}
for
0
≤
λ
≤
1
0 \le \lambda \le 1
and
0
≤
λ
≤
1
/
2
0 \le \lambda \le 1/2
, respectively. To prove these results, we prove that the Lebesgue constants of Hermite–Fejér interpolation operators for the Stieltjes polynomials
E
n
+
1
(
λ
)
E_{n+1}^{(\lambda )}
and the product
E
n
+
1
(
λ
)
P
n
(
λ
)
E_{n+1}^{(\lambda )}P_n^{(\lambda )}
are optimal, that is, the Lebesgue constants
‖
H
n
+
1
‖
∞
(
0
≤
λ
≤
1
)
\|H_{n+1}\|_{\infty }(0 \le \lambda \le 1)
and
‖
H
2
n
+
1
‖
∞
(
0
≤
λ
≤
1
/
2
)
\|{\mathcal H}_{2n+1}\|_{\infty } (0 \le \lambda \le 1/2)
have optimal order
O
(
1
)
O(1)
. In the case of the Hermite–Fejér interpolation polynomials
H
2
n
+
1
[
⋅
]
{\mathcal H}_{2n+1}[\cdot ]
for
1
/
2
>
λ
≤
1
1/2 > \lambda \le 1
, we prove weighted uniform convergence. Moreover, we give some convergence theorems of Hermite–Fejér and Hermite interpolation polynomials for
0
≤
λ
≤
1
0 \le \lambda \le 1
in weighted
L
p
L_p
norms.