A representation of functions as integrals of a kernel
ψ
(
t
;
x
)
\psi (t;x)
, which was introduced by Studden, with respect to functions of bounded variation in
[
0
,
∞
)
[0,\infty )
is obtained whenever the functions satisfy some conditions involving the differential operators
(
d
/
d
t
)
{
f
(
t
)
/
w
i
(
t
)
}
,
i
=
0
,
1
,
2
,
…
(d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots
. The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in
(
0
,
∞
)
(0,\infty )
. Some approximation operators for the approximation of continuous functions in
[
0
,
∞
)
[0,\infty )
arise naturally and are introduced; some sequence-to-function summability methods are also introduced.