We prove that if a weakly even sequence
{
c
k
:
k
=
0
,
±
1
,
…
}
\{ {c_k}:k = 0, \pm 1, \ldots \}
of complex numbers is such that for some
p
>
1
p > 1
we have
\[
∑
m
=
1
∞
2
m
/
q
(
∑
k
=
2
m
−
1
2
m
−
1
|
Δ
(
c
k
+
c
−
k
)
|
p
)
1
/
p
>
∞
,
1
p
+
1
q
=
1
,
\sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} > \infty ,\frac {1}{p} + \frac {1}{q} = 1,
\]
then the symmetric partial sums of the trigonometric series
(
∗
)
∑
k
=
−
∞
∞
c
k
e
i
k
x
( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}}
converge pointwise, except possibly at
x
=
0
(
mod
2
π
)
x = 0(\operatorname {mod} 2\pi )
, to a Lebesgue integrable function,
(
∗
)
( * )
is the Fourier series of its sum, and series
(
∗
)
( * )
converges in
L
1
(
−
π
,
π
)
{L^1}( - \pi ,\pi )
-norm if and only if
lim
|
k
|
→
∞
c
k
ln
|
k
|
=
0
{\lim _{|k| \to \infty }}{c_k}\ln |k| = 0
. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].