Let
{
c
(
n
)
}
\{ c(n)\}
be a complex null sequence such that for some integer
m
≥
1
m \geq 1
and some
p
∈
(
1
,
2
]
p \in (1,2]
\[
∑
|
n
|
>
∞
|
Δ
m
c
(
n
)
|
p
>
∞
and
∑
n
=
1
∞
|
Δ
(
c
(
n
)
−
c
(
−
n
)
)
|
lg
n
>
∞
.
\sum \limits _{|n| > \infty } {|{\Delta ^m}c(n){|^p} > \infty \quad {\text {and}}\quad \sum \limits _{n = 1}^\infty {|\Delta (c(n) - c( - n))|\lg n > \infty .} }
\]
It is shown that the series
\[
(
∗
)
∑
|
n
|
>
∞
c
(
n
)
e
int
,
t
∈
T
=
R
2
π
Z
( * )\quad \sum \limits _{|n| > \infty } {c(n)} {e^{\operatorname {int} }},\quad t \in T = \frac {\mathbb {R}}{{2\pi \mathbb {Z}}}
\]
converges a.e. and that the well-known condition
C
w
{C_w}
of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that
n
Δ
c
(
n
)
=
0
(
1
)
,
|
n
|
→
∞
n\Delta c(n) = 0(1),\quad |n| \to \infty
, implies that the condition
C
w
{C_w}
is equivalent to the de la Vallee Poussin summability of partial sums
{
S
n
(
c
)
}
\{ {S_n}(c)\}
as conjectured in [8].