# Korovkin-Type Theorems for ModularΨ-A-Statistical Convergence

Authors: Carlo Bardaro; Antonio Boccuto; Kamil Demirci; Ilaria Mantellini; Sevda Orhan
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We deal with a new type of statistical convergence for double sequences, calledΨ-A-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.


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## Body

## 1. Introduction

The classical Bohman-Korovkin theorem (see [1–4]) establishes the uniform convergence in the space C
(
[
a
,
b
]
) of all continuous real functions defined on the interval [
a
,
b
], for a sequence (
T
n
)
n of positive linear operators acting on C
(
[
a
,
b
]
), assuming the convergence only on the test functions 1, x, and x
2. For functions belonging to other functional spaces, the problem to extend the Korovkin theorem is much more difficult and, in general, it holds for functions belonging to a suitable subspace. We quote here [5–9] for the Korovkin theorem in L
p-spaces, [10, 11] for an extension to Orlicz spaces, [12–19] for general modular spaces, and [20, 21] for a comprehensive survey about these topics. In recent years, general approaches to the Korovkin theorem were studied, especially in the direction of statistical convergence, which was first introduced independently by Fast and Steinhaus (see [22, 23]). Another classical paper on the emergence of statistical convergence is [24], and then this kind of convergence was developed by many authors (see, e.g., [17, 25–31]). In [32], we have introduced a more general notion of statistical convergence for double sequences of positive linear operators (
T
i
,
j
)
i
,
j, named “triangular A-statistical convergence,” and we have obtained a version of the Korovkin theorem in the space C
(
D
) of all continuous real functions defined in a compact subset D
⊂
R
2.This paper continues and develops the theory introduced in [18, 19, 32, 33]. Here, we consider the more general case of the Ψ-A-statistical convergence, which essentially considers suitable submatrices of the infinite matrix A
=
(
a
i
,
j
)
i
,
j (see Section 2), defined by means of a suitable function Ψ
:
N
×
N
→
R. The triangular A-statistical convergence is defined by the particular choice Ψ
(
i
,
j
)
=
i
-
j. The general definition using an arbitrary function Ψ was already announced in [32]. However, here, we consider the general case when the operators T
i
,
j, i
,
j
∈
N, are acting on an abstract modular function space L
ρ
(
G
) generated by a modular functional ρ and G is a locally compact topological space of finite Borel measure, endowed with a uniform structure. We obtain, in a suitable subspace of L
ρ
(
G
), a Korovkin theorem and also we study certain quantitative versions, under suitable assumptions.Our general approach enables one to unify various extensions of the Korovkin theorem, by choosing suitably the functionΨ, the matrix A, and the modular functional ρ. In particular, corresponding results in L
p, Orlicz, and Musielak-Orlicz spaces are obtained. In Section 4, we also discuss an extension to nonpositive operators, using an approach introduced in [25] and used also in [12]. Finally, some examples are given, in the setting of Mellin-type convolution operators, which are extensively studied in recent years by many authors (see, e.g., [15, 34–41]).


## 2. Preliminaries

LetA
=
(
a
i
,
j
)
i
,
j be a two-dimensional infinite matrix. Given a double sequence x
=
(
x
i
,
j
)
i
,
j of real numbers, set
(1)
A
x
i
:
=
∑
j
=
1
∞
‍
a
i
,
j
x
i
,
j
,
provided that the series is convergent. We say that A isregular if it maps every convergent sequence into a convergent sequence with the same limit. We now recall the Silverman-Toeplitz conditions, which are a characterization of regular two-dimensional matrix transformations (see also [25, 42, 43]):(i)
A
=
sup
⁡
i
∈
N
∑
j
=
1
∞
‍
a
i
,
j
<
∞;
(ii)
lim
⁡
i
a
i
,
j
=
0 for each j
∈
N;
(iii)
lim
⁡
i
∑
j
=
1
∞
‍
a
i
,
j
=
1.LetA
=
(
a
i
,
j
)
i
,
j be a nonnegative regular matrix, Ψ
:
N
×
N
→
R a fixed function, K
⊂
N
2, and K
i
:
=
{
j
∈
N: (
i
,
j
)
∈
K, Ψ
(
i
,
j
)
≥
0
}, and let K
i denote the cardinality of K
i, for each i
∈
N. The Ψ-A-density of K is defined by
(2)
δ
A
Ψ
K
:
=
lim
⁡
i
∑
j
∈
K
i
‍
a
i
,
j
,
provided that the limit on the right-hand side exists in R.It is not difficult to check that theΨ-A-density satisfies the following properties:(i)
ifH
1
⊂
H
2, then δ
A
Ψ
(
H
1
)
≤
δ
A
Ψ
(
H
2
);
(ii)
δ
A
Ψ
(
N
2
∖
K
)
=
1
-
δ
A
Ψ
(
K
), provided that K has a Ψ-A density.Definition 1.
LetA
=
(
a
i
,
j
)
i
,
j be a nonnegative regular summability matrix. The double sequence x
=
(
x
i
,
j
)
i
,
j is said to be Ψ-A-statistically convergent to L if for any ε
>
0 one gets
(3)
lim
⁡
i
∑
j
∈
K
i
(
ε
)
‍
a
i
,
j
=
0
,
where K
i
(
ε
)
=
{
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
x
i
,
j
-
L
≥
ε
}, and we write s
t
A
Ψ-lim
⁡
i
x
i
,
j
=
L.Given a double sequencex
_
=
(
x
i
,
j
)
i
,
j in R, put
(4)
A
x
_
=
a
∈
R
:
δ
A
Ψ
(
{
(
i
,
j
)
∈
N
2
:
x
i
,
j
<
a
}
)
>
0
or
does
not
exist
in
R
δ
A
Ψ
,
B
x
_
=
b
∈
R
:
δ
A
Ψ
(
{
(
i
,
j
)
∈
N
2
:
x
i
,
j
>
b
}
)
>
0
or
does
not
exist
in
R
δ
A
Ψ
,
and define
(5)
s
t
A
Ψ
-
limsup
⁡
i
x
i
,
j
=
sup
⁡
B
x
_
,
if
B
x
_
≠
∅
,
-
∞
,
if
B
x
_
=
∅
,
s
t
A
Ψ
-
liminf
⁡
i
x
i
,
j
=
inf
⁡
A
x
_
,
if
A
x
_
≠
∅
,
+
∞
,
if
A
x
_
=
∅
.
Observe that s
t
A
Ψ
-limsup and s
t
A
Ψ
-liminf satisfy the usual properties of the limit superior and the limit inferior (see also [12, 44]).If we takeΨ
(
i
,
j
)
=
i
-
j, we obtain the notion of triangular A-statistical convergence (see also [32]). Observe that if A
=
C
1 is the Cesàro matrix, defined by setting
(6)
a
i
,
j
:
=
1
i
if
j
≤
i
,
0
otherwise
,
then the Ψ-A-statistical convergence can be considered as a generalized concept of the classical statistical convergence.TheΨ-density δ
Ψ
(
K
) is defined by
(7)
δ
Ψ
K
=
lim
⁡
i
1
i
K
i
.
The double sequence x
=
(
x
i
,
j
)
i
,
j is said to be Ψ-statistically convergent to L if for each ε
>
0 the set K
(
ε
)
:
=
{
(
i
,
j
)
∈
N
2
:
Ψ
(
i
,
j
)
≥
0
,
x
i
,
j
-
L
≥
ε
} has Ψ-density zero, and we write s
t
Ψ
-
lim
⁡
i
x
i
,
j
=
L.We denote bys
t
A
Ψ the set of all Ψ-A-statistically convergent sequences.Remark 2.
(a) It is easy to see that ifΨ
1
≥
Ψ
2, then Ψ
1-A-statistical convergence implies Ψ
2-A-statistical convergence.
Note that, in general, the converse implication is not true. Indeed, letA be the Cesàro matrix, σ
>
0 a fixed real number, and (
x
i
,
j
)
i
,
j any double sequence with x
i
,
j
≥
σ for each i
,
j
∈
N, and put Ψ
1
(
i
,
j
)
=
i
-
j, Ψ
2
(
i
,
j
)
=
i
-
j,  i
,
j
∈
N. We get K
i
′
(
σ
)
:
=
{
j
∈
N
:
j
≤
i
,
x
i
,
j
≥
σ
}
⊆
{
j
∈
N
:
j
≤
i
}, and, hence,
(8)
lim
⁡
i
∑
j
∈
K
i
′
(
σ
)
‍
a
i
,
j
≤
lim
⁡
i
∑
j
≤
i
‍
1
i
=
1
.
So the double sequence (
x
i
,
j
)
i
,
j does not Ψ
1-A-statistically converge to 0. On the other hand, for any double sequence (
x
i
,
j
)
i
,
j, we have K
i
′′
(
σ
)
:
=
{
j
∈
N
:
j
≤
i
,
x
i
,
j
≥
σ
}
⊆
{
j
∈
N
:
j
≤
i
}. So we get
(9)
lim
⁡
i
∑
j
∈
K
i
′′
(
σ
)
‍
a
i
,
j
≤
lim
⁡
i
∑
j
≤
i
‍
1
i
≤
lim
⁡
i
1
i
=
0
,
and thus (
x
i
,
j
)
i
,
j
Ψ
2-A-statistically converges to 0.
(b) Observe that ifA
=
(
a
i
,
j
)
i
,
j is the Cesàro matrix, then (
x
i
,
j
)
i
,
j is a double sequence,Pringsheim convergent to a real number L, namely, such that for every ε
>
0 there is N
∈
N with |
x
i
,
j
-
L
|
<
ε whenever i
,
j
≥
N (see also [45, 46]), then (
x
i
,
j
)
i
,
j is s
t
A
Ψ-convergent to L for every function Ψ
:
N
×
N
→
R. Indeed, if we choose arbitrarily such a Ψ and an ε
>
0, then, for each i
≥
N, we have K
i
(
ε
)
=
{
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
|
x
i
,
j
-
L
|
≥
ε
}
⊂
{
j
∈
N
:
j
≤
N
-
1
}. Thus, we get
(10)
∑
j
∈
K
i
(
ε
)
‍
a
i
,
j
≤
∑
j
=
1
N
-
1
‍
a
i
,
j
=
N
-
1
i
∀
i
≥
N
,
and hence lim
⁡
i
∑
j
∈
K
i
(
ε
)
‍
a
i
,
j
=
0; namely, (
x
i
,
j
)
i
,
j  s
t
A
Ψ-converges to L. Note that the converse, in general, is not true (see also [32, Example 4]).


## 3. The Korovkin Theorem in Modular Spaces

LetG be a locally compact Hausdorff topological space, endowed with a uniform structure U
⊂
2
G
×
G which generates the topology of G (see also [47]). Let B be the σ-algebra of all Borel subsets of G and μ a positive finite regular measure defined on B. We denote by L
0
(
G
) the space of all real-valued μ-measurable functions on G with identification up to sets of measure μ zero, by C
b
(
G
) the space of all real-valued continuous and bounded functions on G, and by C
c
(
G
) the subspace of C
b
(
G
) of all functions with compact support on G.We now recall the notion of modular space (see also [15, 48]).A functionalρ
:
L
0
(
G
)
→
R
0
+
~ is called amodular on L
0
(
G
) if it satisfies the following conditions:(i)
ρ
[
f
]
=
0
⇔
f
=
0   μ-almost everywhere on G;
(ii)
ρ
[
-
f
]
=
ρ
[
f
] for every f
∈
L
0
(
G
);
(iii)
ρ
[
a
f
+
b
g
]
≤
ρ
[
f
]
+
ρ
[
g
] for every f, g
∈
L
0
(
G
) and for each a
≥
0, b
≥
0 with a
+
b
=
1; a modular ρ is said to beconvex if it satisfies conditions (i), (ii),
(iii′)
ρ
[
a
f
+
b
g
]
≤
a
ρ
[
f
]
+
b
ρ
[
g
] for all f
,
g
∈
L
0
(
G
) and for every a, b
≥
0 with a
+
b
=
1.LetQ
≥
1 be a constant. We say that a modular ρ is Q
-quasi-semiconvex if ρ
[
a
f
]
≤
Q
a
ρ
[
Q
f
] for all f
∈
L
0
(
G
), f
≥
0, and 0
<
a
≤
1 (see also [14]).Themodular space L
ρ
(
G
) generated by ρ is defined by
(11)
L
ρ
(
G
)
=
{
f
∈
L
0
(
G
)
:
lim
⁡
λ
→
0
+
ρ
λ
f
=
0
}
,
and thespace of the finite elements of L
ρ
(
G
) is
(12)
E
ρ
(
G
)
=
{
f
∈
L
ρ
(
G
)
:
ρ
λ
f
<
+
∞
∀
λ
>
0
}
.
We will recall the following properties of modulars (see also [15, 48–50]).(a)
A modularρ ismonotone if ρ
f
≤
ρ
g for all f
,
g
∈
L
0
(
G
) with f
≤
g.
(b)
A modularρ isfinite if χ
A (the characteristic function associated with A) belongs to L
ρ
(
G
) whenever A
∈
B with μ
(
A
)
<
+
∞.
(c)
A modularρ isstrongly finite if χ
A belongs to E
ρ
(
G
) for all A
∈
B with μ
(
A
)
<
+
∞.
(d)
A modularρ is said to beabsolutely continuous if there is a positive constant a with the property: for all f
∈
L
0
(
G
) with ρ
[
f
]
<
+
∞,
(i)
for eachε
>
0, there exists a set A
∈
B with μ
(
A
)
<
+
∞ and ρ
[
a
f
χ
G
∖
A
]
≤
ε,
(ii)
for everyε
>
0, there is a δ
>
0 with ρ
[
a
f
χ
B
]
≤
ε for every B
∈
B with μ
(
B
)
<
δ.Example 3 (see [15, 48]).
LetΦ be the set of all continuous nondecreasing functions φ
:
R
0
+
→
R
0
+ with φ
(
0
)
=
0, φ
(
u
)
>
0 for any u
>
0, and lim
⁡
u
→
+
∞
φ
(
u
)
=
+
∞ in the usual sense, and let Φ
~ be the set of all elements of Φ which are convex functions.
For everyφ
∈
Φ (resp., Φ
~), the functional ρ
φ, defined by
(13)
ρ
φ
[
f
]
=
∫
G
‍
φ
(
f
s
)
d
μ
(
s
)
,
f
∈
L
0
(
G
)
,
is a modular (resp., convex) on L
0
(
G
) and
(14)
L
φ
(
G
)
:
=
{
f
∈
L
0
(
G
)
:
ρ
φ
λ
f
<
+
∞
for
some
λ
>
0
}
is theOrlicz space generated by φ.We now define the modular and strong convergences in the context of theΨ-A-statistical convergence (for the classical case and filter convergence, see [12, 15, 36, 37, 48], resp.).A double sequence(
f
i
,
j
)
i
,
j of functions in L
ρ
(
G
) is s
t
A
Ψ-modularly convergent to f
∈
L
ρ
(
G
) if there is a λ
>
0 with
(15)
s
t
A
Ψ
-
lim
⁡
i
ρ
λ
f
i
,
j
-
f
=
0
.
A double sequence (
f
i
,
j
)
i
,
j in L
ρ
(
G
) is s
t
A
Ψ-strongly convergent to f
∈
L
ρ
(
G
) if
(16)
s
t
A
Ψ
-
lim
⁡
i
ρ
λ
f
i
,
j
-
f
=
0
for every λ
>
0.ForΨ
(
i
,
j
)
=
i
-
j, we obtain the corresponding notions for the “triangular” modular and strong convergences.Given a subsetA
⊂
L
ρ
(
G
) and f
∈
L
ρ
(
G
), we say that f
∈
A
¯ (i.e., f is in the modular closure of A) if there is a sequence (
f
n
)
n in A such that (
f
n
)
n is modularly convergent to f in the usual sense.We recall the following.Proposition 4 (see also [51, Theorem 1]).
Letρ be a monotone, strongly finite, and absolutely continuous modular on L
0
(
G
). Then, C
c
(
G
)
¯
=
L
ρ
(
G
) with respect to the modular convergence in the ordinary sense.Throughout this paper, we will consider some kinds of rates of approximation associated with the Korovkin theorem in the context ofΨ-A-statistical convergence, and, for technical reasons, we will sometimes suppose that G
=
(
G
,
d
) is a metric space, satisfying the following property:(
H
*
) for every n
∈
N and s
,
t
∈
G, with s
≠
t, there are n
+
1 points x
i, i
=
1
,
…
,
n
+
1, such that s
=
x
0, t
=
x
n
+
1, and d
(
x
i
,
x
i
+
1
)
≤
1
/
n
d
(
s
,
t
) for each i
=
0
,
…
,
n.Observe that the Euclidean multidimensional spaces endowed with the usual metric fulfil(
H
*
), as well as the space R
Λ endowed with the sup-norm, where Λ is any abstract nonempty set.Forf
∈
C
b
(
G
) and δ
>
0, let
(17)
ω
(
f
;
δ
)
:
=
sup
⁡
{
f
s
-
f
t
:
s
,
t
∈
G
,
d
(
s
,
t
)
≤
δ
}
be the usual modulus of continuity of f. Observe that ω
(
f
;
δ
) is an increasing function of δ, f
s
-
f
t
≤
ω
(
f
;
d
(
s
,
t
)
) for each s, t
∈
G, and ω
(
f
;
δ
)
≤
2
M for every δ, where M
=
sup
⁡
t
∈
G
f
t and
(18)
ω
(
f
;
γ
δ
)
≤
(
1
+
γ
)
ω
(
f
;
δ
)
for every γ, δ
>
0 (see also [21]).We now prove some Korovkin-type theorems with respect to an abstract finite set of test functionse
0
,
…
,
e
m in the context of the Ψ-A-statistical convergence.LetT be a double sequence of linear operators T
i
,
j
:
D
→
L
0
(
G
), i
,
j
∈
N, with C
b
(
G
)
⊂
D
⊂
L
0
(
G
). Here, the set D is the domain of the operators T
i
,
j.We say that the double sequenceT, together with the modular ρ, satisfies theproperty
(
ρ
)
-
(
*
) if there exist a subset X
T
⊂
D
∩
L
ρ
(
G
) with C
b
(
G
)
⊂
X
T and a positive real constant N with T
i
,
j
f
∈
L
ρ
(
G
) for any f
∈
X
T   and i
,
j
∈
N, and s
t
A
Ψ
-
limsup
⁡
i
ρ
[
τ
(
T
i
,
j
f
)
]
≤
N
ρ
[
τ
f
] for every f
∈
X
T and τ
>
0.Some examples of operators satisfying property(
ρ
)-(
*
) can be found in [14].Sete
0
(
t
)
≡
1 for every t
∈
G; let e
r, r
=
1
,
…
,
m, and a
r, r
=
0
,
…
,
m, be functions in C
b
(
G
). Put
(19)
P
s
(
t
)
:
=
∑
r
=
0
m
‍
a
r
(
s
)
e
r
(
t
)
,
s
,
t
∈
G
,
and assume that (P1)
P
s
(
s
)
=
0 for all s
∈
G,
(P2)
for every neighborhoodU
∈
U there is a positive real number η with P
s
(
t
)
≥
η whenever s
,
t
∈
G, (
s
,
t
)
∉
U (see also [12]).From now on, we will assume thate
r
∈
L
ρ
(
G
), r
=
0,1
,
…
,
m. This happens, for example, when G is an open bounded subset of R
n or, more generally, a space of finite measure μ.We now prove the following.Theorem 5.
Letρ be a strongly finite, monotone, and Q-quasi-semiconvex modular. Assume that e
r and a
r, r
=
0
,
…
,
m, satisfy (P1) and (P2). Let T
i
,
j, i
,
j
∈
N, be a double sequence of positive linear operators with property (
ρ
)-(
*
). If (
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-modularly convergent to e
r in L
ρ
(
G
) for each r
=
0
,
…
,
m, then (
T
i
,
j
f
)
i
,
j is s
t
A
Ψ-modularly convergent to f in L
ρ
(
G
) for every f
∈
C
c
(
G
).
If(
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-strongly convergent to e
r, r
=
0
,
…
,
m, in L
ρ
(
G
), then (
T
i
,
j
f
)
i
,
j is s
t
A
Ψ-strongly convergent to f in L
ρ
(
G
) for every f
∈
C
c
(
G
).Proof.
Letf
∈
C
c
(
G
). Since G is endowed with the uniformity U, f is uniformly continuous and bounded on G. Fix arbitrarily ε
∈
]
0,1
[. By uniform continuity of f, there is an element U
∈
U with f
s
-
f
t
≤
ε whenever s
,
t
∈
G, (
s
,
t
)
∈
U.
For eachs, t
∈
G let P
s
(
t
) be as in (19), and, in correspondence with U, let η
>
0 satisfy condition (P2). If M
=
sup
⁡
t
∈
G
f
t, then we get
(20)
f
s
-
f
t
≤
2
M
≤
2
M
η
P
s
t
whenever
s
,
t
∈
G
,
(
s
,
t
)
∉
U
.
In any case,
(21)
f
s
-
f
t
≤
ε
+
2
M
η
P
s
t
∀
s
,
t
∈
G
;
namely,
(22)
-
ε
-
2
M
η
P
s
t
≤
f
s
-
f
t
≤
ε
+
2
M
η
P
s
t
∀
s
,
t
∈
G
.
Since T
i
,
j is a positive linear operator, by applying T
i
,
j to (22), for every i
,
j
∈
N and s
∈
G, we get
(23)
-
ε
T
i
,
j
e
0
s
-
2
M
η
T
i
,
j
P
s
s
≤
f
(
s
)
(
T
i
,
j
e
0
)
(
s
)
-
(
T
i
,
j
f
)
(
s
)
≤
ε
(
T
i
,
j
e
0
)
(
s
)
+
2
M
η
(
T
i
,
j
P
s
)
(
s
)
,
and, hence,
(24)
T
i
,
j
f
s
-
f
s
≤
T
i
,
j
f
s
-
f
s
T
i
,
j
e
0
s
+
f
s
T
i
,
j
e
0
s
-
f
s
≤
ε
(
T
i
,
j
e
0
)
(
s
)
+
2
M
η
(
T
i
,
j
P
s
)
(
s
)
+
M
T
i
,
j
e
0
s
-
e
0
s
.
Let γ
>
0. By applying the modular ρ, from (24) for any i
,
j
∈
N, we get
(25)
ρ
γ
T
i
,
j
f
-
f
≤
ρ
[
3
γ
ε
(
T
i
,
j
e
0
)
]
+
ρ
[
3
γ
M
(
T
i
,
j
e
0
-
e
0
)
]
+
ρ
[
6
γ
M
η
(
T
i
,
j
P
(
·
)
)
(
·
)
]
=
J
1
+
J
2
+
J
3
.
So, to prove the theorem, it is sufficient to find a positive real number γ, with s
t
A
Ψ
-
lim
⁡
i
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
=
0. Indeed, let λ
>
0 be with s
t
A
Ψ
-
lim
⁡
i
ρ
[
λ
(
T
i
,
j
e
r
-
e
r
)
]
=
0 for each r
=
0
,
…
,
m: such a λ, by hypothesis, does exist. Choose an N
>
0 with |
a
r
(
s
)
|
≤
N for each r
=
0
,
…
,
m and s
∈
G, and let γ
>
0 be with max
⁡
{
3
γ
M
,
6
γ
M
/
η
(
m
+
1
)
N
}
≤
λ. Taking into account (P1), for every i
,
j
∈
N, we get
(26)
J
3
=
ρ
6
γ
M
η
T
i
,
j
P
·
·
=
ρ
6
γ
M
η
T
i
,
j
P
·
·
-
P
·
·
≤
∑
r
=
0
m
‍
ρ
6
γ
M
η
m
+
1
N
T
i
,
j
e
r
-
e
r
≤
∑
r
=
0
m
‍
ρ
λ
T
i
,
j
e
r
-
e
r
.
So, s
t
A
Ψ
-
lim
⁡
i
J
3
=
0. Moreover, thanks to the choice of λ and γ, it is not difficult to see that s
t
A
Ψ
-
lim
⁡
i
J
2
=
0.
Asρ is Q-quasi-semiconvex and ε
∈
]
0,1
[, then
(27)
ρ
[
3
γ
ε
e
0
]
≤
Q
ε
ρ
[
3
γ
Q
e
0
]
.
By taking the limit superior and thanks to (
ρ
)-(
*
), from (25) and (27), we obtain
(28)
0
≤
s
t
A
Ψ
-
limsup
⁡
i
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
≤
s
t
A
Ψ
-
limsup
⁡
i
ρ
3
γ
ε
T
i
,
j
e
0
≤
N
ρ
3
γ
ε
e
0
≤
N
Q
ε
ρ
[
3
γ
Q
e
0
]
.
From (28), by arbitrariness of ε and strong finiteness of ρ, we get
(29)
s
t
A
Ψ
-
limsup
⁡
i
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
=
0
,
and hence s
t
A
Ψ
-
lim
⁡
i
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
=
0. Thus, the double sequence T
i
,
j
f, i
,
j
∈
N, is s
t
A
Ψ-modularly convergent to f in L
ρ
(
G
).
The proof of the last part of the theorem is analogous.We now give the main Korovkin-type theorem.Theorem 6.
Letρ be a monotone, strongly finite, absolutely continuous, and Q-quasi-semiconvex modular on L
0
(
G
) and T
i
,
j, i
,
j
∈
N, a double sequence of positive linear operators fulfilling (
ρ
)-(
*
). If (
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-strongly convergent to e
r, r
=
0
,
…
,
m, in L
ρ
(
G
), then (
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-modularly convergent to f in L
ρ
(
G
) for every f
∈
L
ρ
(
G
)
∩
D with f
-
C
b
(
G
)
⊂
X
T, where D and X
T are as before.Proof.
Letf
∈
L
ρ
(
G
)
∩
D with f
-
C
b
(
G
)
⊂
X
T. By Proposition 4, there are a λ
>
0 and a sequence f
k, k
∈
N, in C
c
(
G
) with ρ
[
3
λ
f
]
<
+
∞ and lim
⁡
k
ρ
[
3
λ
(
f
k
-
f
)
]
=
0 in the usual sense.
Fix arbitrarilyε
>
0 and pick a positive integer k
¯ with
(30)
ρ
[
3
λ
(
f
k
¯
-
f
)
]
≤
ε
.
For every i
,
j
∈
N, we have
(31)
ρ
[
λ
(
T
i
,
j
f
-
f
)
]
≤
ρ
[
3
λ
(
T
i
,
j
f
-
T
i
,
j
f
k
¯
)
]
+
ρ
[
3
λ
(
T
i
,
j
f
k
¯
-
f
k
¯
)
]
+
ρ
[
3
λ
(
f
k
¯
-
f
)
]
.
By virtue of Theorem 5, we get
(32)
0
=
s
t
A
Ψ
-
lim
⁡
i
ρ
[
3
λ
(
T
i
,
j
f
k
¯
-
f
k
¯
)
]
=
s
t
A
Ψ
-
limsup
⁡
i
ρ
[
3
λ
(
T
i
,
j
f
k
¯
-
f
k
¯
)
]
.
By property (
ρ
)-(
*
), we find an N
>
0 with
(33)
s
t
A
Ψ
-
lim
⁡
i
ρ
[
3
λ
(
T
i
,
j
f
-
T
i
,
j
f
k
¯
)
]
≤
N
ρ
[
3
λ
(
f
-
f
k
¯
)
]
≤
N
ε
.
From (30), (31), (32), (33), and subadditivity of the s
t
A
Ψ
-limsup, we obtain
(34)
0
≤
s
t
A
Ψ
-
limsup
⁡
i
ρ
[
λ
(
T
i
,
j
f
-
f
)
]
≤
ε
(
N
+
1
)
.
From (34) and arbitrariness of ε
>
0, it follows that
(35)
s
t
A
Ψ
-
limsup
⁡
i
ρ
[
λ
(
T
i
,
j
f
-
f
)
]
=
0
,
and so s
t
A
Ψ
-
lim
⁡
i
ρ
[
λ
(
T
i
,
j
f
-
f
)
]
=
0, that is, the assertion.Remark 7.
(a) Note that, in Theorem6, in general, it is not possible to obtain s
t
A
Ψ-strong convergence unless the modular ρ satisfies the Δ
2 condition; namely, there is a positive real number c
0 with ρ
[
2
f
]
≤
c
0
ρ
[
f
] for any f
∈
L
0
(
G
) (for the classical case, see also [48]).
(b) By a similar technique, we can prove an analogous result in the spaceC
2
π
(
R
) of all continuous real-valued functions on R of period 2
π, by the homeomorphic identification of C
2
π
(
[
0,2
π
]
) with C
(
S
1
).


## 4. Rates ofΨ-A-Statistical Convergence

We present some estimates of rates of triangularA-statistical convergence for Korovkin-type theorems in the setting of modular convergence, extending earlier results proved in [32].Definition 8.
LetA
=
(
a
i
,
j
)
i
,
j be a nonnegative regular summability matrix and (
α
i
)
i a nonincreasing sequence of positive real numbers. A double sequence x
=
(
x
i
,
j
)
i
,
j is Ψ-A
-statistically convergent to a number L
with the rate of
o
k
(
α
i
) if, for every ε
>
0,
(36)
lim
⁡
i
∑
j
∈
M
i
(
ε
)
‍
a
i
,
j
=
0
,
where
(37)
M
i
(
ε
)
:
=
{
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
x
i
,
j
-
L
≥
ε
α
j
}
.
In this case, we write
(38)
x
i
,
j
-
L
=
s
t
A
Ψ
-
o
k
α
i
as
i
⟶
+
∞
.Definition 9.
LetA
=
(
a
i
,
j
)
i
,
j and (
α
i
)
i be as in Definition 8. Then, a double sequence x
=
(
x
i
,
j
)
i
,
j is Ψ-A-statistically bounded with the rate of O
k
(
α
i
) if there is an M
>
0 with
(39)
lim
⁡
i
∑
j
∈
N
i
(
M
)
‍
a
i
,
j
=
0
,
where
(40)
N
i
(
M
)
:
=
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
x
i
,
j
≥
M
α
j
.
In this case, we write
(41)
x
i
,
j
=
s
t
A
Ψ
-
O
k
α
i
as
i
⟶
+
∞
.We now recall the following results (see also [32, Lemmas 3 and 4]).Lemma 10.
Letx
i
,
j
i
,
j and y
i
,
j
i
,
j be two double sequences, A
=
(
a
i
,
j
)
i
,
j a nonnegative regular summability matrix, and α
i
i and β
i
i two positive nonincreasing sequences. If x
i
,
j
-
L
1
=
s
t
A
Ψ
-
o
k
(
α
i
) and y
i
,
j
-
L
2
=
s
t
A
Ψ
-
o
k
(
β
i
) as i
→
+
∞, then we have(i)
(
x
i
,
j
-
L
1
)
±
(
y
i
,
j
-
L
2
)
=
st
A
Ψ
-
o
k
(
γ
i
) as i
→
+
∞, where γ
i
:
=
max
⁡
α
i
,
β
i for each i
∈
N,
(ii)
λ
(
x
i
,
j
-
L
1
)
=
s
t
A
Ψ
-
o
k
(
α
i
) as i
→
+
∞ for any real number λ.Furthermore, similar results hold when the symbol o is replaced by O.Now, we are in position to prove the following.Theorem 11.
LetT
i
,
j
i
,
j, A
=
(
a
i
,
j
)
i
,
j, and e
0 be as above, let ρ be a monotone and strongly finite modular, and suppose that (
G
,
d
) is a metric space satisfying condition (H
*). Let (
α
i
)
i and (
β
i
) be two nonincreasing sequences of strictly positive real numbers, and put γ
i
:
=
max
⁡
{
α
i
,
β
i
} for each i
∈
N. For each s
,
t
∈
G, set ψ
(
s
)
(
t
)
=
d
(
s
,
t
), and for every f
∈
C
c
(
G
) and i
,
j
∈
N let δ
i
,
j
=
T
i
,
j
(
ψ
), where the symbol · denotes the sup-norm and the supremum is taken with respect to the support of f. Furthermore, let there exist γ
>
0 with(i)
ρ
[
γ
(
T
i
,
j
e
0
-
e
0
)
]
=
s
t
A
Ψ
-
o
k
(
α
i
) as i
→
+
∞,
(ii)
ρ
[
γ
ω
(
f
;
δ
i
,
j
)
]
=
s
t
A
Ψ
-
o
k
(
β
i
) as i
→
+
∞.
Then for everyf
∈
C
c
(
G
) we get ρ
[
γ
(
T
i
,
j
f
-
f
)
]
=
s
t
A
Ψ
-
o
k
(
γ
i
) as i
→
+
∞.
Furthermore, similar results hold when the symbolo is replaced by O.Proof.
Letf
∈
C
c
(
G
), M
=
sup
⁡
t
∈
G
f
t. Observe that ω
(
f
;
δ
)
≤
2
M for each δ
>
0. Using the properties of the modulus of continuity, we get
(42)
f
s
-
f
t
≤
ω
(
f
;
d
(
s
,
t
)
)
≤
1
+
d
s
,
t
δ
ω
f
;
δ
for every δ
>
0 and s
,
t
∈
G. Moreover, observe that, since f
∈
C
c
(
G
), the quantities δ
i
,
j, i
,
j
∈
N, are well-defined. Indeed the support of f is (totally) bounded, and so sup
⁡
s
,
t
∈
G
d
(
s
,
t
)
<
+
∞. By applying T
i
,
j, i
,
j
∈
N, we find a positive real number E
i
,
j (depending on i, j, and f, but not on s and t) with sup
⁡
s
,
t
∈
G
T
i
,
j
(
d
(
s
,
t
)
)
≤
E
i
,
j
T
i
,
j
(
e
0
). From this, it follows that δ
i
,
j
∈
R for each i
,
j
∈
N.
Let nowδ
=
δ
i
,
j. By applying T
i
,
j, keeping fixed s and letting t vary in G, and taking into account linearity and monotonicity, from (42), we obtain
(43)
T
i
,
j
f
s
-
f
s
≤
T
i
,
j
(
1
+
d
(
s
,
t
)
δ
ω
(
f
;
δ
)
)
≤
ω
f
;
δ
T
i
,
j
e
0
s
-
e
0
s
+
ω
(
f
;
δ
)
δ
T
i
,
j
(
ψ
)
(
s
)
+
ω
(
f
;
δ
)
e
0
(
s
)
+
M
T
i
,
j
e
0
s
-
e
0
s
≤
4
M
(
T
i
,
j
e
0
s
-
e
0
s
+
ω
(
f
;
δ
)
)
for each s
∈
G. Let now γ
>
0. By applying the modular ρ, from (43), we obtain
(44)
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
≤
ρ
[
8
γ
M
(
T
i
,
j
e
0
-
e
0
)
]
+
ρ
[
8
γ
M
ω
(
f
;
δ
)
]
.
For each i
∈
N and ε
>
0, set
(45)
U
i
,
ε
:
=
{
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
ρ
[
γ
(
T
i
,
j
f
-
f
)
]
≥
ε
γ
j
}
,
U
i
,
ε
1
≔
ε
α
j
2
j
∈
N
:
Ψ
i
,
j
≥
0
,
h
h
h
h
h
l
h
ρ
8
γ
M
T
i
,
j
e
0
-
e
0
≥
ε
α
j
2
,
U
i
,
ε
2
:
=
j
∈
N
:
Ψ
(
i
,
j
)
≥
0
,
ρ
[
8
γ
M
ω
(
f
;
δ
)
]
≥
ε
β
j
2
.
From (44), it follows that
(46)
U
i
,
ε
⊂
U
i
,
ε
1
∪
U
i
,
ε
2
,
and hence
(47)
∑
j
∈
U
i
,
ε
‍
a
i
,
j
≤
∑
j
∈
U
i
,
ε
1
‍
a
i
,
j
+
∑
j
∈
U
i
,
ε
2
‍
a
i
,
j
.
Letting i tend to +
∞, we obtain
(48)
lim
⁡
i
∑
j
∈
U
i
,
ε
‍
a
i
,
j
=
0
,
that is, the assertion.


## 5. An Extension to Nonpositive Operators

One can ask whether it is possible in the Korovkin-type theorems to relax the positivity condition on the linear operators involved. We now give some positive answers also in the context ofΨ-A-statistical convergence, extending earlier results of  [12, 25], proved in the settings of A-convergence and filter convergence, respectively.LetA
=
(
a
i
,
j
)
i
,
j be a nonnegative regular matrix, I a bounded interval of R, and C
2
(
I
) (resp., C
b
2
(
I
)) the space of all functions defined on I (resp., bounded and) continuous together with their first and second derivatives, C
+
:
=
{
f
∈
C
b
2
(
I
)
:
f
≥
0
}
,
C
+
2
:
=
{
f
∈
C
b
2
(
I
)
:
f
′′
≥
0
}.Lete
r, r
=
1
,
…
,
m, and a
r, r
=
0
,
…
,
m, be functions in C
b
2
(
I
) and let P
s
(
t
), s
,
t
∈
I, be as in (19), and suppose that P
s
(
t
) satisfies the above conditions (P1) and (P2) and the following condition:(P3)
there is a positive real constantC
0 with P
s
′′
(
t
)
≥
C
0 for all s
,
t
∈
I (here the second derivative is intended with respect to t).Some examples in which properties (P1), (P2), and (P3) are satisfied can be found in [36].We now prove the following Korovkin-type theorem for not necessarily positive linear operators, in the setting ofΨ-A-statistical convergence.Theorem 12.
LetA
=
(
a
i
,
j
)
i
,
j be a nonnegative regular summability matrix, let ρ be as in Theorem 5, and let e
r, a
r, r
=
0
,
…
,
m, and P
s
(
t
), s
,
t
∈
I, satisfy properties (P1), (P2), and (P3). Assume that T
i
,
j, i
,
j
∈
N, is a double sequence of linear operators, satisfying property (
ρ
)-(
*
), and that δ
A
Ψ
{
(
i
,
j
)
∈
N
2
:
Ψ
(
i
,
j
)
≥
0
,
T
i
,
j
(
C
+
∩
C
+
2
)
⊂
C
+
}
=
1. If (
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-modularly convergent to e
r, r
=
0
,
…
,
m, in L
ρ
(
I
), then (
T
i
,
j
f
)
i
,
j is s
t
A
Ψ-modularly convergent to f in L
ρ
(
I
) for each f
∈
C
b
2
(
I
).
If(
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-strongly convergent to e
r, r
=
0
,
…
,
m, in L
ρ
(
I
), then (
T
i
,
j
f
)
i
,
j is s
t
A
Ψ-strongly convergent to f in L
ρ
(
I
) for every f
∈
C
b
2
(
I
).
Furthermore, ifρ is absolutely continuous and (
T
i
,
j
e
r
)
i
,
j is s
t
A
Ψ-strongly convergent to e
r, r
=
0
,
…
,
m, in L
ρ
(
I
), then (
T
i
,
j
f
)
i
,
j is s
t
A
Ψ-modularly convergent to f in L
ρ
(
I
) for every f
∈
L
ρ
(
I
)
∩
D with f
-
C
b
(
I
)
⊂
X
T.Proof.
Letf
∈
C
b
2
(
I
). Note that f is uniformly continuous and bounded on I. Fix arbitrarily ε
∈
]
0,1
[. By uniform continuity of f there is a δ
>
0 with |
f
(
s
)
-
f
(
t
)
|
≤
ε for all s
,
t
∈
I, |
s
-
t
|
≤
δ.
LetP
s
(
t
), s
,
t
∈
I, be as in (19), and let η
>
0 be associated with δ, satisfying (P2). By arguing analogously as in the proof of Theorem 5, for every β
≥
1 and s
,
t
∈
I, we get
(49)
-
ε
-
2
M
β
η
P
s
(
t
)
≤
f
(
s
)
-
f
(
t
)
≤
ε
+
2
M
β
η
P
s
(
t
)
,
where M
=
sup
⁡
t
∈
I
|
f
(
t
)
|. From (49), it follows that
(50)
h
1
,
β
(
t
)
:
=
2
M
β
η
P
s
(
t
)
+
ε
+
f
(
t
)
-
f
(
s
)
≥
0
,
h
2
,
β
(
t
)
:
=
2
M
β
η
P
s
(
t
)
+
ε
-
f
(
t
)
+
f
(
s
)
≥
0
for all β
≥
1 and s
,
t
∈
I. Let C
0 satisfy (P3). For each t
∈
I, we have
(51)
h
1
,
β
′′
(
t
)
≥
2
M
β
C
0
η
+
f
′′
(
t
)
,
h
2
,
β
′′
(
t
)
≥
2
M
β
C
0
η
-
f
′′
(
t
)
.
Since f
′′ is bounded on I, we can choose β
≥
1 in such a way that h
1
,
β
′′
(
t
)
≥
0 and h
2
,
β
′′
(
t
)
≥
0 for each t
∈
I. Thus, h
1
,
β
,
h
2
,
β
∈
C
+
∩
C
+
2. Let K
0
:
=
{
(
i
,
j
)
∈
N
2
:
Ψ
(
i
,
j
)
≥
0
,
T
i
,
j
(
C
+
∩
C
+
2
)
⊂
C
+
}: by hypothesis, δ
A
Ψ
(
K
0
)
=
1 and
(52)
T
i
,
j
(
h
j
,
β
)
(
s
)
≥
0
∀
(
i
,
j
)
∈
K
0
,
s
∈
I
,
j
=
1,2
.
From (50), (52), and linearity of T
i
,
j, for each (
i
,
j
)
∈
K
0 and s
∈
I, we have
(53)
2
M
β
η
(
T
i
,
j
P
s
)
(
s
)
+
ε
(
T
i
,
j
e
0
)
(
s
)
+
(
T
i
,
j
f
)
(
s
)
-
f
(
s
)
(
T
i
,
j
e
0
)
(
s
)
≥
0
,
2
M
β
η
(
T
i
,
j
P
s
)
(
s
)
+
ε
(
T
i
,
j
e
0
)
(
s
)
-
(
T
i
,
j
f
)
(
s
)
+
f
(
s
)
(
T
i
,
j
e
0
)
(
s
)
≥
0
,
and, hence,
(54)
-
ε
(
T
i
,
j
e
0
)
(
s
)
-
2
M
β
η
(
T
i
,
j
P
s
)
(
s
)
≤
f
(
s
)
(
T
i
,
j
e
0
)
(
s
)
-
(
T
i
,
j
f
)
(
s
)
≤
ε
(
T
i
,
j
e
0
)
(
s
)
+
2
M
β
η
(
T
i
,
j
P
s
)
(
s
)
.
By arguing analogously as in the proof of Theorem 5, using the modular ρ and taking into account that δ
A
Ψ
(
K
0
)
=
1, we get the assertion of the first part. The other parts can be proved by proceeding similarly as in the proofs of Theorems 5 and 6.


## 6. Some Application to Mellin-Type Operators

In this section, we give some application of Theorem6 to bivariate moment-type operators (see also [37, 52]). We will consider the case of an Orlicz space L
φ
(
G
), with φ
∈
Φ
~ and G
=
[
0,1
]
2
⊂
R
2.For every(
x
1
,
x
2
)
∈
[
0,1
]
2, let e
0
(
x
1
,
x
2
)
=
a
3
(
x
1
,
x
2
)
=
1, e
1
(
x
1
,
x
2
)
=
x
1, e
2
(
x
1
,
x
2
)
=
x
2, e
3
(
x
1
,
x
2
)
=
a
0
(
x
1
,
x
2
)
=
x
1
2
+
x
2
2, a
1
(
x
1
,
x
2
)
=
-
2
x
1, and a
2
(
x
1
,
x
2
)
=
-
2
x
2. For each i
,
j
∈
N, set A
i
,
j
=
[
1
/
i
,
1
]
×
[
1
/
j
,
1
],
(55)
c
i
,
j
=
∫
0
1
‍
∫
0
1
‍
t
1
t
2
t
1
2
+
t
2
2
i
+
j
χ
A
i
,
j
t
1
,
t
2
d
t
1
d
t
2
,
d
i
,
j
=
1
c
i
,
j
.
For every i
,
j
∈
N and t
1
,
t
2
∈
[
0,1
], define K
i
,
j
(
t
1
,
t
2
)
=
d
i
,
j
t
1
t
2
(
t
1
2
+
t
2
2
)
i
+
j
χ
A
i
,
j
(
t
1
,
t
2
). Moreover, let C
1 be the Cesàro matrix. Finally, let us consider the double sequences of operators defined by
(56)
M
i
,
j
f
x
1
,
x
2
=
∫
0
1
‍
∫
0
1
‍
K
i
,
j
t
1
,
t
2
f
t
1
x
1
,
t
2
x
2
d
t
1
d
t
2
,
f
∈
C
0,1
2
,
i
,
j
∈
N
.
We will prove that the M
i
,
j’s satisfy Theorem 6. We begin with the following.Lemma 13 (see [32, Lemma 1]).
For everyi
,
j
≥
2, we get
(57)
d
i
,
j
≤
i
+
j
+
1
2
2
i
+
j
-
3
,
where d
i
,
j is as in (55).Lemma 14 (see also [32, Lemma 2]).
For everyδ
∈
]
0,1
[, there exists a
δ
∈
]
0,1
[ with
(58)
∫
‍
∫
A
i
,
j
∖
U
δ
‍
K
i
,
j
(
t
1
,
t
2
)
d
t
1
d
t
2
≤
8
π
(
i
+
j
+
1
)
a
δ
i
+
j
+
1
for every i
,
j
∈
N, where U
δ denotes the ball with center (
1,1
) and radius δ.Lemma 15.
For everyi
,
j
≥
2, one gets ∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
/
t
1
t
2
d
t
1
d
t
2
≤
32.Proof.
Leti
,
j
≥
2. Then, taking into account Lemma 13, we have
(59)
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
t
1
t
2
d
t
1
d
t
2
=
d
i
,
j
∫
0
1
‍
∫
0
1
‍
t
1
2
+
t
2
2
i
+
j
d
t
1
d
t
2
≤
d
i
,
j
∫
0
1
‍
d
t
1
∫
0
1
‍
t
1
+
t
2
i
+
j
d
t
2
=
d
i
,
j
i
+
j
+
1
∫
0
1
‍
t
1
+
1
i
+
j
+
1
-
t
1
i
+
j
+
1
d
t
1
≤
d
i
,
j
i
+
j
+
1
∫
0
1
‍
t
1
+
1
i
+
j
+
1
d
t
1
=
d
i
,
j
(
i
+
j
+
1
)
(
i
+
j
+
2
)
(
2
i
+
j
+
2
-
1
)
≤
i
+
j
+
1
i
+
j
+
2
·
2
i
+
j
+
2
2
i
+
j
-
3
≤
32
.Lemma 16.
For eachi
,
j
≥
2 and f
∈
L
φ
(
[
0,1
]
2
), one gets ρ
φ
[
M
i
,
j
f
]
≤
32
ρ
φ
[
f
].Proof.
By virtue of the Jensen inequality and the Fubini theorem, we have(60)
ρ
φ
M
i
,
j
f
≤
∫
0
1
‍
∫
0
1
‍
K
i
,
j
φ
f
t
1
x
1
,
t
2
x
2
d
x
1
d
x
2
d
t
1
d
t
2
≤
ρ
φ
[
f
]
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
t
1
t
2
d
t
1
d
t
2
≤
32
ρ
φ
[
f
]
.
We now show that the hypotheses of Theorem 6 are satisfied. First of all, note that, by construction, we get M
i
,
j
(
f
0
)
(
x
1
,
x
2
)
=
f
0
(
x
1
,
x
2
)
=
1 for each i
,
j
∈
N and x
1
,
x
2
∈
[
0,1
].
Fix nowε
∈
]
0
,
1
/
4
[ and let U
ε
⊂
R
2 be the ball of center (
1,1
) and radius ε. For every i
,
j
∈
N, x
1
,
x
2
∈
[
0,1
], and r
=
1,2, we get
(61)
M
i
,
j
f
r
x
1
,
x
2
-
f
r
x
1
,
x
2
≤
x
r
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
(
1
-
t
r
)
d
t
1
d
t
2
≤
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
(
1
-
t
r
)
d
t
1
d
t
2
=
∫
‍
∫
A
i
,
j
∩
U
ε
‍
K
i
,
j
t
1
,
t
2
1
-
t
r
d
t
1
d
t
2
+
∫
‍
∫
A
i
,
j
∖
U
ε
‍
K
i
,
j
(
t
1
,
t
2
)
(
1
-
t
r
)
d
t
1
d
t
2
≤
2
∫
A
i
,
j
∖
U
ε
‍
K
i
,
j
t
1
,
t
2
d
t
1
d
t
2
+
π
2
ε
2
≤
2
∫
A
i
,
j
∖
U
ε
‍
K
i
,
j
t
1
,
t
2
d
t
1
d
t
2
+
ε
2
≤
8
π
(
i
+
j
+
1
)
a
ε
i
+
j
+
1
+
ε
2
.
From this and arbitrariness of ε, it follows that for r
=
1,2 and for every ε
>
0 there exists N
∈
N with
(62)
M
i
,
j
e
r
x
1
,
x
2
-
f
r
x
1
,
x
2
≤
ε
whenever
i
,
j
≥
N
,
x
1
,
x
2
∈
[
0,1
]
.
Now, fix arbitrarily λ
>
0. From (62), since 0
<
ε
<
1 and taking into account convexity of the function φ, we get
(63)
ρ
φ
λ
M
i
,
j
e
r
-
e
r
≤
ε
ρ
φ
λ
for
each
i
,
j
≥
N
,
x
1
,
x
2
∈
[
0,1
]
.
From (63) and Remark 2(b), it follows that s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
e
r
-
e
r
)
]
=
0, r
=
1,2. Arguing analogously as above, considering for r
=
1,2 the estimate
(64)
M
i
,
j
f
r
2
x
1
,
x
2
-
f
r
2
x
1
,
x
2
≤
x
r
2
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
(
1
-
t
r
2
)
d
t
1
d
t
2
,
and taking into account Remark 2(b), it is possible to check that
(65)
s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
e
r
2
-
e
r
2
)
]
=
0
,
r
=
1,2
,
and so, by linearity, we get s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
e
3
-
e
3
)
]
=
0. Furthermore, note that, from Lemma 16, it follows that the condition ρ-(
*
) is satisfied, with X
T
=
L
φ
(
[
0,1
]
2
) (see also [14, Section 3]). Thus, by virtue of Theorem 6, for every f
∈
L
φ
(
[
0,1
]
2
), there is λ
>
0 with s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
f
)
-
f
)
]
=
0.We now consider a direct extension to the bivariate case of the classical one-dimensional moment kernel (see also [37, 52]).For everyi
,
j
∈
N and t
1
,
t
2
∈
[
0,1
], let K
i
,
j
(
t
1
,
t
2
)
=
(
i
+
1
)
(
j
+
1
)
t
1
i
t
2
j, and for f
∈
C
(
[
0,1
]
2
) and x
1
,
x
2
∈
[
0,1
] set
(66)
(
M
i
,
j
f
)
(
x
1
,
x
2
)
=
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
f
(
t
1
x
1
,
t
2
x
2
)
d
t
1
d
t
2
.
We have
(67)
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
d
t
1
d
t
2
=
(
i
+
1
)
∫
0
1
‍
t
1
i
d
t
1
(
j
+
1
)
∫
0
1
‍
t
2
j
d
t
2
=
1
,
and so (
M
i
,
j
f
0
)
(
x
1
,
x
2
)
=
f
0
(
x
1
,
x
2
)
=
1 for all i
,
j
∈
N and x
1
,
x
2
∈
[
0,1
]. Furthermore, we get
(68)
M
i
,
j
f
1
x
1
,
x
2
-
f
1
x
1
,
x
2
≤
∫
0
1
‍
∫
0
1
‍
K
i
,
j
(
t
1
,
t
2
)
(
1
-
t
1
)
d
t
1
d
t
2
=
(
i
+
1
)
(
j
+
1
)
∫
0
1
‍
t
1
i
(
1
-
t
1
)
d
t
1
∫
0
1
‍
t
2
j
d
t
2
=
(
i
+
1
)
∫
0
1
‍
t
1
i
d
t
1
-
(
i
+
1
)
∫
0
1
‍
t
1
i
+
1
d
t
1
=
1
-
i
+
1
i
+
2
=
1
i
+
2
for i
,
j
∈
N and x
1
,
x
2
∈
[
0,1
]. Analogously, it is possible to see that
(69)
M
i
,
j
f
2
x
1
,
x
2
-
f
2
x
1
,
x
2
≤
1
j
+
2
,
M
i
,
j
f
1
2
x
1
,
x
2
-
f
1
2
x
1
,
x
2
≤
2
i
+
3
,
M
i
,
j
f
2
2
x
1
,
x
2
-
f
2
2
x
1
,
x
2
≤
2
j
+
3
whenever i
,
j
∈
N and x
1
,
x
2
∈
[
0,1
]. So we obtain s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
e
r
-
e
r
)
]
=
0 for each λ
>
0 and r
=
0,1
,
2,3. Thus, all the hypotheses of Theorem 6 are satisfied, and, hence, for every f
∈
L
φ
(
[
0,1
]
2
) there is λ
>
0 with s
t
C
1
Ψ
-
lim
⁡
i
ρ
φ
[
λ
(
M
i
,
j
f
-
f
)
]
=
0.



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*Source: 160401-2015-01-31.xml*