Let
l
∞
{l_\infty }
denote the Banach space of bounded sequences,
σ
\sigma
an injection of the set of positive integers into itself having no finite orbits, and
T
T
the operator defined on
l
∞
{l_\infty }
by
T
y
(
n
)
=
y
(
σ
n
)
Ty\left ( n \right ) = y\left ( {\sigma n} \right )
. A positive linear functional
L
\mathcal {L}
with
‖
L
‖
=
1
\left \| \mathcal {L} \right \| = 1
, is called a
σ
\sigma
-mean if
L
(
y
)
=
L
(
T
y
)
\mathcal {L}\left ( y \right ) = \mathcal {L}\left ( {{T_y}} \right )
for all
y
y
in
l
∞
{l_\infty }
. A sequence
y
y
is said to be
σ
\sigma
-convergent, denoted
y
∈
V
σ
y \in {V_\sigma }
, if
L
(
y
)
\mathcal {L}\left ( y \right )
takes the same value, called
σ
−
lim
y
\sigma - \lim y
, for all
σ
\sigma
-means
L
\mathcal {L}
. P. Schaefer [6] gave necessary and sufficient conditions on a matrix
A
A
to ensure that
A
(
c
)
⊂
V
σ
A\left ( c \right ) \subset {V_\sigma }
, where
c
c
is the space of convergent sequences, and additional conditions ensuring that
σ
−
lim
A
y
=
lim
y
\sigma - \lim Ay = \lim y
for all
y
∈
c
y \in c
, denoting the class of matrices satisfying these conditions by
(
c
,
V
σ
)
1
{\left ( {c,{V_\sigma }} \right )_1}
and calling them the
σ
\sigma
-regular matrices. In this paper, we use such matrices to find the sum of a sequence of Walsh functions.