Let
z
n
k
=
e
i
t
n
k
{z_{nk}} = {e^{i{t_{nk}}}}
,
0
≤
t
n
0
>
⋯
>
t
n
n
>
2
π
0 \leq {t_{n0}} > \cdots > {t_{nn}} > 2\pi
,
f
f
a function in the disc algebra
A
A
, and
μ
n
=
max
{
|
t
n
k
−
2
k
π
/
(
n
+
1
)
|
:
0
≤
k
≤
n
}
{\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\}
. Denote by
L
n
(
f
;
⋅
)
{L_n}(f;\; \cdot )
the polynomial of degree
n
n
that agrees with
f
f
at
{
z
n
k
:
k
=
0
,
…
,
n
}
\{ {z_{nk}}:k = 0, \ldots ,n\}
. In this paper, we prove that for every
p
p
,
0
>
p
>
∞
0 > p > \infty
, there exists a
δ
p
>
0
{\delta _p} > 0
, such that
|
|
L
n
(
f
;
⋅
)
−
f
|
|
p
=
O
(
ω
(
f
;
1
n
)
)
||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n}))
whenever
μ
n
≤
δ
p
/
(
n
+
1
)
{\mu _n} \leq {\delta _p}/(n + 1)
. It must be emphasized that
δ
p
{\delta _p}
necessarily depends on
p
p
, in the sense that there exists a family
{
z
n
k
:
k
=
0
,
…
,
n
}
\{ {z_{nk}}:k = 0, \ldots ,n\}
with
μ
n
=
δ
2
/
(
n
+
1
)
{\mu _n} = {\delta _2}/(n + 1)
and such that
|
|
L
n
(
f
;
⋅
)
−
f
|
|
2
=
O
(
ω
(
f
;
1
n
)
)
||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n}))
for all
f
∈
A
f \in A
, but
sup
{
|
|
L
n
(
f
;
⋅
)
|
|
p
:
f
∈
A
,
|
|
f
|
|
∞
=
1
}
\sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\}
diverges for sufficiently large values of
p
p
. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for
{
z
n
k
}
\{ {z_{nk}}\}
.