Abstract
We use wavelets to define the Kantorovich variant of
q
-Baskakov type operators, and for
1
≤
p
<
∞
, we study the
L
p
-approximation. Let
ξ
be any positive constant and
Ψ
k
(
x
)
be any continuous derivative function such that
∫
R
x
s
Ψ
k
(
x
)
d
q
x
=
0
where
0
≤
s
≤
k
,
k
∈
N
,
0
<
q
<
1
.
For all
Ψ
∈
L
∞
(
R
)
suppose the following conditions hold: (i) a finite positive
ξ
exits with the property
sup
Ψ
⊂
[
0
,
ξ
]
,
(ii) its first
k
moments vanish: For
1
≤
s
≤
k
,
k
∈
N
, we have
∫
R
t
s
Ψ
(
t
)
d
q
t
=
0
and
∫
R
Ψ
(
t
)
d
q
t
=
1
. Then in the sense of Haar basis for
0
<
q
<
1
,
the
q
-
analogue of Baskakov–Kantorovich type wavelets operators are defined by
S
r
,
s
,
q
g
(
x
)
=
[
r
]
q
∑
s
=
0
∞
q
s
-
1
B
r
,
s
,
q
(
x
)
∫
R
g
t
Ψ
q
s
-
1
[
r
]
q
t
-
[
s
]
q
d
q
t
.