Journal of Inequalities in Pure and
Applied Mathematics
ON GLOBAL APPROXIMATION PROPERTIES OF ABSTRACT
INTEGRAL OPERATORS IN ORLICZ SPACES
AND APPLICATIONS
volume 6, issue 4, article 123,
2005.
CARLO BARDARO AND ILARIA MANTELLINI
Department of Mathematics and Informatics
University of Perugia
Via Vanvitelli 1, 06123 Perugia, Italy.
Received 25 August, 2005;
accepted 01 September, 2005.
Communicated by: S.S. Dragomir
EMail: bardaro@unipg.it
EMail: mantell@dipmat.unipg.it
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2000
Victoria University
ISSN (electronic): 1443-5756
253-05
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Abstract
In this paper we study approximation properties for the class of general integral
operators of the form
Z
(Tw f)(s) =
Kw (s, t, f(t))dµHw (t) s ∈ G, w > 0
Hw
where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of
closed subsets of G with suitable properties and, for every w > 0, µHw is a
regular measure on Hw . We give pointwise, uniform and modular convergence
theorems in abstract modular spaces and we apply the results to some kinds of
discrete operators including the sampling type series.
2000 Mathematics Subject Classification: 41A25, 41A35, 47G10, 46E30.
Key words: Modular approximation, nonlinear integral operators, regular families,
singularity.
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Convergence in Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4
Applications to Discrete Operators . . . . . . . . . . . . . . . . . . . . . . 22
References
On Global Approximation
Properties of Abstract Integral
Operators in Orlicz Spaces and
Applications
Carlo Bardaro and Ilaria Mantellini
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
1.
Introduction
The aim of this paper is to give a general and unifying treatment of convergence properties in function spaces for families of integral or discrete operators, including the classical linear or nonlinear integral operators of convolution
type acting on function spaces defined on topological groups, Urysohn type
operators, discrete operators of sampling type, or the classical Bernstein and
Szász-Mirak’jan operators etc. Our objective is to find a common root for the
convergence properties of all the above operators.
In [4] we have studied approximation properties for families of nonlinear
integral operators of the form
Z
(Tw f )(s) =
Kw (s, t, f (t))dµHw (t), s ∈ G, w > 0,
On Global Approximation
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Operators in Orlicz Spaces and
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Carlo Bardaro and Ilaria Mantellini
Hw
where G is a locally compact Hausdorff
topological space, (Hw )w>0 is a net
S
of closed subsets of G such that w>0 Hw = G and, for every w > 0, µHw is
a regular and σ-finite measure on Hw . For these operators we have obtained
some local convergence theorems in the general setting of modular spaces. Locality represents a restriction of applicability of the theory and so in this paper
we furnish global approximation results by using a notion of uniform modular
integrability for families of functions introduced in [5] for Urysohn type operators and suitably modified according to our general frame. We will work in the
setting of Orlicz spaces, but using some natural generalization of the uniform
integrability, some technical tool and an approach similar to [4], we can obtain
corresponding results also in abstract modular spaces. As a consequence we
obtain results in Lp spaces, when the generating function of the Orlicz space is
given by ϕ(u) = |u|p .
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
By specializing G and Hw , we obtain several classes of operators. For example, if G = Hw , and µw = µG , for every w > 0, µG being a fixed σ−finite
and regular measure on G, we obtain a class of Urysohn type operators and so,
in turn, all the classical families of linear or nonlinear integral operators of convolution type (when G is a topological group and µG is its Haar measure). Our
main theorems (Theorems 3.4 and 3.6) imply the well-known approximation
results in Orlicz spaces (or, in particular, in Lp −spaces), see [13], [24]).
Here we consider, as examples, very interesting families of operators of the
discrete type (see Section 4), in order to explain the nature of the general conditions which we used. In particular, we apply these results to linear discrete
operators of the form ([26], [14], [15], [16], [7], [22] and [6])
n
X
(Sw f )(s) =
K(ws − n)f
, s ∈ R, w > 0
w
n∈Z
and
(Sew f )(s) =
X
n∈Z
Gn,w (s)f
tn
w
s ∈ R, w > 0,
where (tn )n∈N is a separate sequence of real numbers (i.e. |tn − tn−1 | ≥ δ for
a fixed δ > 0), (see [21], [17], [3]) and to their nonlinear versions (see e.g. [8],
[27], [23], [6], [3], [4] and [1]). In these instances, our general convergence
theorems imply modular convergence in Orlicz spaces for the generalized sampling operators (in their linear or nonlinear forms), when the function f satisfies
some generalized concepts of bounded variation. As far we know, this is a new
result about generalized sampling operators (also in Lp −case). Finally, we also
discuss another well-known discrete operator, namely the Szász-Mirak’jan operator and we obtain, as an application, a convergence result in Orlicz space.
On Global Approximation
Properties of Abstract Integral
Operators in Orlicz Spaces and
Applications
Carlo Bardaro and Ilaria Mantellini
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
2.
Notations and Definitions
Let G be a locally compact Hausdorff topological space, provided with its
family of Borel sets B of G. Let µG be a regular and σ−finite measure defined
on B. We will assume that the topology of G is uniformizable, i.e. there is a
uniform structure U ⊂ G × G which generates the topology of G (see [28]).
For every U ∈ U, we put Us = {t ∈ G : (s, t) ∈ U }. By local compactness,
we will assume that for every s ∈ G, the base {Us : U ∈ U} contains
compact sets. We will denote by X(G) the space of all real-valued measurable
functions f : G → R provided with equality a.e., by C(G) the subspace of all
bounded and continuous functions and by Cc (G) the subspace of all continuous
functions with compact support.
Let us recall that a function f : G → R is uniformly continuous on G if for
every ε > 0 there is U ∈ U such that |f (t)−f (s)| < ε for every s ∈ G, t ∈ Us .
Let now (Hw )w>0 be a net of nonempty closed subsets of G such that
G = ∪w>0 Hw .
For every w > 0 we will denote by µw a regular and σ-finite measure
on Hw , defined on the Borel σ-algebra generated by the family {A ∩ Hw :
A open subset of G}.
Let L be the class of all the families of globally measurable functions
L = (Lw )w>0 , Lw : G × Hw → R+
0 such that for every w > 0 the sections
1
Lw (·, t) and Lw (s, ·) belong to L (G) for every t ∈ Hw and to L1 (Hw ) for
every s ∈ G respectively.
+
Let Ψ be the class of all functions ψ : R+
0 → R0 such that ψ is a continuous
(nondecreasing) function and ψ(0) = 0, ψ(u) > 0 for u > 0. Let Ξ =
(ψw )w>0 ⊂ Ψ be a family of functions such that the following two assumptions
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hold
1. (ψw )w>0 is equicontinuous at u = 0,
2. for every u ≥ 0 the net (ψw (u))w>0 is bounded.
We denote by KΞ the class of all families of functions K = (Kw )w>0 , where
Kw : G × Hw × R → R, such that the following conditions hold
(K.1) for any w > 0, Kw (·, ·, u) is measurable on G × Hw for every u ∈ R and
Kw (s, t, 0) = 0, for every (s, t) ∈ G × Hw .
(K.2) K = (Kw )w>0 is (L, Ξ)−Lipschitz i.e. there are L = (Lw )w>0 ∈ L and
a constant D > 0 such that
Z
0 < βw (s) :=
Lw (s, t)dµw (t) ≤ D
Hw
for all s ∈ G, w > 0 and
|Kw (s, t, u) − Kw (s, t, v)| ≤ Lw (s, t)ψw (|u − v|)
for every s ∈ G, t ∈ Hw and u, v ∈ R.
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For a given K = (Kw )w>0 ∈ KΞ we will take into consideration the following family of nonlinear integral operators T = (Tw )w>0 given by
Z
(Tw f )(s) =
Kw (s, t, f (t))dµw (t) s ∈ G, w > 0
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Hw
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
where f ∈ Dom T = ∩w>0 Dom Tw ; here Dom Tw is the subset of X(G) on
which Tw f is well defined as a µG -measurable function of s ∈ G.
We will say that K = (Kw )w>0 ∈ KΞ is singular if the following assumptions hold
1) There is a net (ζw )w>0 of positive real numbers such that for every w >
0 and t ∈ Hw we have
Z
Lw (s, t)dµG (s) ≤ ζw ≤ D.
G
2) For every s ∈ G and for every U ∈ U we have
Z
lim
Lw (s, t)dµw (t) = 0.
w→+∞
Hw
We will say that K is uniformly singular if conditions 2) and 3) are replaced
by the following ones
2’) for every U ∈ U we have
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lim
w→+∞
Carlo Bardaro and Ilaria Mantellini
Hw \Us
3) For every s ∈ G and for every u ∈ R we have
Z
lim
Kw (s, t, u)dµw (t) = u.
w→+∞
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Lw (s, t)dµw (t) = 0
Hw \Us
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uniformly with respect to s ∈ G,
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
3’) we have
Z
lim
w→+∞
Kw (s, t, u)dµw (t) = u,
Hw
uniformly with respect to s ∈ G and u ∈ C, where C is any compact
subset of R \ {0}.
For the above concepts see [4].
We have the following
Theorem 2.1. Let K = (Kw )w>0 ∈ KΞ . Then L∞ (G) ⊂ Dom T and for every
w > 0 Tw f ∈ L∞ (G), whenever f ∈ L∞ (G).
Proof. We obtain easily the measurability of Kw (s, ·, f (·)) and Tw f, with f ∈
L∞ (G). Moreover
Z
|(Tw f )(s)| ≤
Lw (s, t)ψw (|f (t)|)dµw (t) ≤ ψw (kf k∞ )D ≤ M 0 D,
Hw
0
being M = supw>0 (ψw (kf k∞ )) and so the assertion follows.
As to the pointwise convergence, in [4] we have proved the following result
Theorem 2.2. Let f ∈ L∞ (G) and K = (Kw )w>0 ∈ KΞ be singular. Then
Tw f → f pointwise at every continuity point of f. Moreover if K is uniformly
singular, then kTw f − f k∞ → 0 as w → +∞, whenever f ∈ C(G) is
uniformly continuous.
On Global Approximation
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3.
Convergence in Orlicz Spaces
+
Let Φ be the class of all functions ϕ : R+
0 → R0 such that
i) ϕ is continuous and non decreasing,
ii) ϕ(0) = 0, ϕ(u) > 0 for u > 0 and limu→+∞ ϕ(u) = +∞.
e the subspace of Φ whose elements are convex
Moreover we denote by Φ
functions.
For ϕ ∈ Φ, we define the functional
Z
ϕ
%G (f ) =
ϕ(|f (s)|)dµG (s)
G
On Global Approximation
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Carlo Bardaro and Ilaria Mantellini
for every f ∈ X(G).
As it is well known, %ϕG is a modular on X(G) and the subspace
Lϕ (G) = {f ∈ X(G) : %ϕG (λf ) < +∞ for some λ > 0}
e then %ϕ is a convex modular. The
is the Orlicz space generated by ϕ. If ϕ ∈ Φ,
G
subspace of Lϕ (G), defined by
E ϕ (G) = {f ∈ X(G) : %ϕG (λf ) < +∞ for every λ > 0},
is called the space of finite elements of Lϕ (G). For example, every bounded
function with compact support belongs to E ϕ (G). For these concepts see [24].
Moreover, given an arbitrary η ∈ Φ, we will denote by %ηHw the modular
Z
η
%Hw (f ) =
η(|f (t)|)dµw (t)
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Hw
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
for any f ∈ X(G).
Here by Lη (Hw ) we denote the space of all functions f ∈ X(G) such that
the restriction f|Hw is an element of the Orlicz space generated by the modular
%ηHw .
In order to establish a convergence result in Orlicz spaces, we consider the
following notion of convergence (see [24]). We say that a sequence (fw )w>0 ⊂
Lϕ (G) is modularly convergent to f ∈ Lϕ (G) if there is λ > 0 such that
lim %ϕG [λ(fw − f )] = 0.
w→+∞
This notion of convergence induces a topology on Lϕ (G), called modular topology.
In the following we will need a link between the modular %ϕHw and the family
of functions Ξ = (ψw )w>0 used in the Lipschitz condition of (Kw )w>0 . Given
a function η ∈ Φ, considering the modular %ηHw , we will say that the triple
(%ϕHw , ψw , %ηHw ) is properly directed if the following condition holds: there
exists a net (cw )w>0 , with cw → 0 as w → +∞, for which for every λ ∈
]0, 1[ there exists Cλ ∈]0, 1[ with
Z
Z
(3.1)
ϕ(Cλ ψw (|f (t)|))dµw (t) ≤
η(λ|f (t)|)dµw (t) + cw
Hw
Hw
for every function f ∈ X(G).
We will need the following definition: given
(Lw (·, t))t∈Hw ,w>0 ,
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
we will say that this family satisfies property (∗), if for every compact C ⊂
G there exist a compact subset B ⊂ G, M > 0 and β > 0 such that
Z
M
(3.2)
Lw (s, t)dµG (s) ≤ β
w
G\B
for every t ∈ C ∩ Hw and sufficiently large w > 0.
3.1.
Examples
1. As a first example, let G = Hw = R, for every w > 0, provided with
the Lebesgue measure. Let us consider an approximate identity (see [13])
e w )w>0 on G = R, such that for every δ > 0
(K
Z
β
e w (s)|ds = M > 0,
lim w
|K
w→+∞
|t|>δ
for some β > 0. Then we can prove that the family of kernels (Kw )w>0 , with
Kw : R × R × R → R defined by
e w (s − t)u,
Kw (s, t, u) = K
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e w (s −
satisfies the singularity assumptions and the family Lw (s, t) = |K
t)| satisfies property (∗). For example the above holds for every family
e w )w>0 of functions K
e w with supports contained in a fixed interval of
(K
the form [−a, a].
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
2. Let G = Hw = R for every w > 0, provided with Lebesgue measure.
Let us consider the family of functions
Lw (s, t) =
2w
ew(s+t)
· 2ws
,
π e + e2wt
for s, t ∈ R and w ≥ 1. Then for every t ∈ R and r > 0 we have
Z
2
Lw (s, t)ds = 1 + (arctan e−wr − arctan ewr ).
π
R\[t−r,t+r]
Now it is easy to show that (3.2) holds for every β > 0. This is an example
in non-convolution case (see [4]).
On Global Approximation
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3. More interesting cases, which involve discrete operators, will be given in
Section 4. These examples can be easily modified in order to get conditions for which the assumption (∗) holds.
Let η ∈ Φ be fixed. For a given constant P > 0 and a net (γw )w>0 of
positive numbers, we denote by ΥHw (P, (γw )w>0 ) the class
ΥHw (P, (γw )w>0 )
= {f ∈ Lη (G) : lim sup γw %ηHw (λf ) ≤ P %ηG (λf ) for every λ > 0}.
w→+∞
Let K = (Kw )w>0 ∈ KΞ be singular and let (ζw )w>0 be the net in assumption
1) of singularity.
In the following we will work with the class ΥHw (P, (ζw )w>0 ) and we will
assume that ΥHw (P, (ζw )w>0 ) contains a nonempty subspace F ⊂ Cc (G). For
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a fixed modular functional %ηG on X(G), we will denote by F η the modular
closure in Lη (G) of F. Thus f ∈ F η iff there exists a sequence (fn )n∈N ⊂
F such that (fn )n∈N converges modularly to f with respect to the modular
%ηG . We have the following
e η ∈ Φ, Ξ = (ψw )w>0 ⊂ Ψ and let K =
Theorem 3.1. Let ϕ ∈ Φ,
(Kw )w>0 ∈ KΞ be singular. Let f ∈ F and let C be the support of f. Let
us assume that wβ ζw → +∞ as w → +∞ and (3.1) holds. If the family
(Lw (·, t))t∈Hw ,w>0 satisfies property (∗), then there exists a constant α > 0, independent of f, such that following properties hold
i) for every ε > 0 there is a compact set B ⊂ G such that
On Global Approximation
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%ϕG [α(Tw f )χG\B ] < ε
for sufficiently large w > 0,
ii) for every sequence (Bk )k∈N ⊂ B, with Bk+1 ⊂ Bk and µG (Bk ) → 0, we
have
lim %ϕG [α(Tw f )χBk ] = 0,
k→+∞
uniformly with respect to w > 0.
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Proof. Let λ > 0 be fixed and let Cλ be the constant in (3.1). Let α > 0 be a
constant such that αD ≤ Cλ . Using the Jensen inequality and the Fubini-Tonelli
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
Theorem, we have for every measurable subset B ⊂ G
Z
ϕ
%G [α(Tw f )χG\B ] =
ϕ(α|(Tw f )(s)|)dµG (s)
G\B
Z
Z
1
ϕ(αDψw (|f (t)|))Lw (s, t)dµw (t) dµG (s)
≤
D G\B Hw ∩C
Z
Z
1
=
ϕ(αDψw (|f (t)|))
Lw (s, t)dµG (s) dµw (t).
D Hw ∩C
G\B
By property (∗) we can consider a compact subset B ⊂ G such that
Z
M
Lw (s, t)dµG (s) ≤ β ,
w
G\B
for every t ∈ Hw ∩ C, with β > 0. So we obtain
Z
M
ϕ
ϕ(αDψw (|f (t)|))dµw (t).
%G [α(Tw f )χG\B ] ≤
Dwβ Hw
Using condition (3.1), we obtain
%ϕG [α(Tw f )χG\B ]
Z
M
η(λ|f (t)|)dµw (t) + cw
≤
Dwβ Hw
Z
M cw
M
η(λ|f (t)|)dµw (t) +
=
β
Dw Hw
Dwβ
Z
M
M cw
=
ζw
η(λ|f (t)|)dµw (t) +
.
β
Dw ζw
Dwβ
Hw
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Letting w → +∞, we have
lim sup %ϕG [α(Tw f )χG\B ]
w→+∞
M
ζw
≤ lim sup
β
w→+∞ Dw ζw
Z
η(λ|f (t)|)dµw (t)
Hw
and, taking into account that f ∈ ΥHw (P, (ζw )w>0 ) and that every function
f ∈ F belongs to E η (G), we have that
lim sup %ϕG [α(Tw f )χG\B ] = 0
w→+∞
and so i) follows. Now we can prove ii). Let (Bk )k∈N ⊂ B be a sequence of
measurable sets with Bk+1 ⊂ Bk and limk→+∞ µG (Bk ) = 0. Then, considering
a constant α such that αD ≤ 1, we obtain
Z
ϕ
ϕ(α|(Tw f )(s)|χBk (s))dµG (s)
%G [α(Tw f )χBk ] =
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G
≤ ϕ(ψw (kf k∞ ))µG (Bk ).
Since (ψw (kf k∞ ))w>0 is bounded we have easily the assertion.
Using Theorems 2.2 and 3.1, we obtain the following
e η ∈ Φ, Ξ = (ψw )w>0 ⊂ Ψ and K = (Kw )w>0 ∈
Theorem 3.2. Let ϕ ∈ Φ,
KΞ be singular. Let us assume that wβ ζw → +∞ as w → +∞ and (3.1)
holds. If the family (Lw (·, t))t∈Hw ,w>0 satisfies property (∗), then there exists
a constant α > 0 such that
lim %ϕG [α(Tw f − f )] = 0,
w→+∞
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for every f ∈ F.
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Proof. Let λ > 0 be fixed and let α > 0 be such that 2αD ≤ Cλ . From
Theorem 2.2, Tw f converges pointwise to f so by continuity of ϕ we have
that ϕ(α|Tw f − f |) tends to zero pointwise too. Moreover by Theorem 3.1 the
integrals
Z
ϕ(α|(Tw f )(s) − f (s)|)dµG (s)
(·)
are equiabsolutely continuous. Thus the assertion follows from the Vitali convergence theorem.
In order to give a modular convergence theorem for functions f ∈ F ϕ+η (G)∩
Dom T, we prove the following continuity result for the family (Tw )w>0 .
e η ∈ Φ and Ξ = (ψw )w>0 ⊂ Ψ be such that
Theorem 3.3. Let ϕ ∈ Φ,
ϕ
η
the triple (%Hw , ψw , %Hw ) is properly directed. Let K = (Kw )w>0 ∈ KΞ be
singular. If f, g ∈ X(G) ∩ Dom T, such that f − g ∈ ΥHw (P, (ζw )w>0 ) then,
given λ > 0, there exists α > 0 such that
lim sup %ϕG [α(Tw f
w→+∞
P
− Tw g)] ≤ %ηG [λ(f − g)].
D
Proof. Let λ > 0 be fixed and let α be a positive constant such that αD ≤
Cλ , with Cλ being the constant in (3.1). By using the Jensen inequality and
the Fubini-Tonelli theorem, we have
%ϕG [α(Tw f
− Tw g)]
Z
ϕ(α|(Tw f )(s) − (Tw g)(s)|)dµG (s)
=
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
Z
≤
ϕ α
Lw (s, t)ψw (|f (t) − g(t)|)dµw (t) dµG (s)
G
Hw
Z
Z
1
≤
ϕ(Cλ ψw (|f (t) − g(t)|))
Lw (s, t)dµG (s) dµw (t)
D Hw
G
Z
ζw
≤
ϕ(Cλ ψw (|f (t) − g(t)|))dµw (t)
D Hw
Z
ζw
cw ζw
≤
η(λ|f (t) − g(t)|)dµw (t) +
D Hw
D
cw ζw
ζw
= %ηHw [λ(f − g)] +
.
D
D
Passing to the limsup, taking into account that f − g ∈ ΥHw (P, (ζw )w>0 ), we
obtain
P
lim sup %ϕG [α(Tw f − Tw g)] ≤ %ηG (λ(f − g))
D
w→+∞
and so the assertion follows.
Z
Now we are ready to prove the main theorem of this section.
e η ∈ Φ and Ξ = (ψw )w>0 ⊂ Ψ be such that
Theorem 3.4. Let ϕ ∈ Φ,
ϕ
η
the triple (%Hw , ψw , %Hw ) is properly directed. Let K = (Kw )w>0 ∈ KΞ be
singular. Let us assume that wβ ζw → +∞ as w → +∞, and the family (Lw (·, t))t∈Hw ,w>0 satisfies property (∗). Then, for every f ∈ Dom T ∩
F η+ϕ , such that f − F ⊂ ΥHw (P, (ζw )w>0 ), for some P > 0, there exists a
constant α > 0 such that
lim %ϕG [α(Tw f − f )] = 0.
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w→+∞
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
Proof. Let f ∈ F η+ϕ ∩ Dom T, such that f − F ⊂ ΥHw (P, (ζw )w>0 ). Then
there is a constant λ0 ∈]0, 1[ and a sequence (fn )n∈N ⊂ F such that for every
ε > 0 there exists n ∈ N with
0
%ϕ+η
G [λ (fn − f )] < ε
for every n ≥ n. Let us fix now n and a constant α > 0 such that 3α ≤ λ0 , for
which Theorems 3.2 and 3.3 are satisfied with 3α. Then we have
%ϕG [α(Tw f −f )] ≤ %ϕG [3α(Tw f −Tw fn )]+%ϕG [3α(Tw fn −fn )]+%ϕG [3α(fn −f )].
By Theorem 3.2 we have
lim %ϕG [3α(Tw fn − fn )] = 0
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w→+∞
and by Theorem 3.3 we have
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lim sup %ϕG [3α(Tw f − Tw fn ) ≤
w→+∞
P η 0
% (λ (f − fn )).
D G
Without loss of generality we can assume that P/D > 1 and so we can write
lim sup %ϕG [α(Tw f − f )] ≤
w→+∞
P η 0
P
P
%G [λ (f − fn )] + %ϕG [λ0 (f − fn )] < ε .
D
D
D
The assertion follows since ε is arbitrary.
Now we describe a different approach to the convergence problem by using
a more general assumption than property (∗). This assumption has been used in
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[5] in connection with Urysohn type operators. However we have to introduce
a “local” boundedness condition on the family of measures (µw )w>0 . Using
the previous notations, we will say that the family (Lw (·, t))t∈Hw ,w>0 satisfies
property (∗∗) if for every compact C ⊂ G and for every ε > 0 there exists a
compact subset B ⊂ G such that
Z
Lw (s, t)dµG (s) < ε,
G\B
for every t ∈ Hw ∩ C and sufficiently large w > 0.
We will say that the family of measures (µw )w>0 is locally bounded if for
every compact C ⊂ G, (µw (C ∩ Hw ))w>0 is a bounded net, i.e. there exists
a constant R = RC > 0 such that µw (C ∩ Hw ) ≤ R, for sufficiently large
w > 0.
As before, let F be a subspace of Cc (G) such that F ⊂ ΥHw (P, (ζw )w>0 ) for
some constant P > 0. We have the following
e Ξ = (ψw )w>0 ⊂ Ψ and let K = (Kw )w>0 ∈
Theorem 3.5. Let ϕ ∈ Φ,
KΞ be singular. Let f ∈ F and let C be the support of f. If the family
(Lw (·, t))t∈Hw ,w>0 satisfies property (∗∗) and the family of measures (µw )w>0 is
locally bounded, then there exists a constant α > 0, independent of f, such that
following properties hold
i) for every ε > 0 there is a compact set B ⊂ G such that
%ϕG [α(Tw f )χG\B ] < ε
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for sufficiently large w > 0,
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
ii) for every sequence (Bk )k∈N ⊂ B, with Bk+1 ⊂ Bk and µG (Bk ) → 0, we
have
lim %ϕG [α(Tw f )χBk ] = 0,
k→+∞
uniformly with respect to w > 0.
Proof. Let α > 0 be such that αD ≤ 1. Let ε > 0 be fixed and let B be the
compact set corresponding to the support C of f in the definition of property
(∗∗). Using the same arguments as in Theorem 3.1, we have, for sufficiently
large w,
Z
Z
1
ϕ
%G [α(Tw f )χG\B ] ≤
ϕ(αDψw (|f (t)|))
Lw (s, t)dµG (s) dµw (t)
D Hw ∩C
G\B
Z
εR
ε
≤
ϕ(ψw (|f (t)|))dµw (t) ≤
ϕ(M 0 ),
D Hw ∩C
D
where M 0 = supw>0 ψw (kf k∞ ) and so i) follows. The proof of ii) is exactly
the same as before.
We remark that Theorem 3.5 still holds for every bounded function with
compact support.
As a consequence we obtain a modular convergence result as in Theorem 3.2.
Finally, using Theorem 3.3, Theorem 3.5 and the Vitali convergence Theorem,
we obtain the restatement of Theorem 3.4
e η ∈ Φ and Ξ = (ψw )w>0 ⊂ Ψ be such that
Theorem 3.6. Let ϕ ∈ Φ,
ϕ
η
the triple (%Hw , ψw , %Hw ) is properly directed. Let K = (Kw )w>0 ∈ KΞ be
singular. Let us assume that the family (Lw (·, t))t∈Hw ,w>0 satisfies property
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
(∗∗) and the family of measures (µw )w>0 is locally bounded. Then, for every
f ∈ Dom T ∩ F η+ϕ , such that f − F ⊂ ΥHw (P, (ζw )w>0 ), for some P >
0, there exists a constant α > 0 such that
lim %ϕG [α(Tw f − f )] = 0.
w→+∞
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4.
Applications to Discrete Operators
Let us consider now the particular case in which G = R and Hw = w1 Z for
every w > 0. We provide G with the Lebesgue measure and for every w >
0 we put µw = w1 µc , where µc is the counting measure. Note that the family
(µw )w>0 is locally bounded. Indeed let C = [−M, M ] be a closed interval in
R and M ∈ N. Then the set C ∩ Hw contains at most 2[M w] + 1 elements.
So µw (C ∩ Hw ) ≤ 2M + 1, for w ≥ 1 and hence the assertion follows.
Let now Γ : R → R be a summable function and such that the following
assumptions hold
P
(L.1) +∞
k=−∞ Γ(u − k) = 1 for every u ∈ R.
P
(L.2) m0 (Γ) := supu∈R +∞
k=−∞ |Γ(u − k)| < +∞.
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Note that from the summability of Γ we deduce the following property
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(L.3) For every ε > 0 there exists M > 0 such that
Z
|Γ(s)|ds < ε.
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|s|>M
Now, for every w > 0, we define
k
k
Γw s,
= wΓ w s −
= wΓ(ws − k),
w
w
s ∈ R, k ∈ Z.
We prove that the family of kernels Kw : G × Hw × R → R, defined by
k
Kw s, , u = wΓ(ws − k)u,
w
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is singular with Lw (s, wk ) = w|Γ(ws − k)|.
Firstly, note that the family (Kw )w>0 satisfies a strong Lipschitz condition
with ψw (u) = u, for every u ≥ 0 and w > 0. Thus assumption (3.1) is satisfied
with η = ϕ and cw = 0, for every w > 0. Moreover
Z
Z
w|Γ(ws − k)|ds =
|Γ(u)|du = kΓk1 < +∞.
R
R
For a fixed s ∈ R and δ > 0, let us put Us = [s − δ, s + δ]. Then
Z
X
k
Lw s,
dµw =
|Γ(ws − k)|.
w
Hw \Us
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Carlo Bardaro and Ilaria Mantellini
By Lemma 1 in [26], assumption (L.2) implies
X
lim
|Γ(u − k)| = 0,
R→+∞
|u−k|>R
Contents
uniformly with respect to u ∈ R and so for a fixed ε > 0 there is Rε > 0 such
that
X
|Γ(u − k)| < ε,
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|u−k|>Rε
uniformly with respect u ∈ R. Let w > 0 such that wδ > Rε for every w ≥
w. Then for w ≥ w,
X
X
|Γ(ws − k)| ≤
|Γ(ws − k)| < ε,
|ws−k|>δw
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|ws−k|>Rε
J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
and so the second condition of singularity in satisfied. The third condition immediately follows from (L.1). Finally we show that property (∗∗) is satisfied
for the family (Lw (·, wk )). Let C = [−δ, δ] be a fixed interval in R and let
f > 0 such
ε > 0 be fixed. Let M be the constant in (L.3) and let us choose M
k
f
that M − δ > M. Since w ∈ [−δ, δ], we only consider k ∈ [−δw, δw]. So we
have, with w > 1,
Z
fw−k
−M
Z
w|Γ(ws − k)|ds =
f
|s|>M
Z
+∞
|Γ(u)|du +
−∞
|Γ(u)|du
fw−k
M
Z
≤
|Γ(u)|du
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|u|>(M
Carlo Bardaro and Ilaria Mantellini
Z
|Γ(u)|du < ε.
<
|u|>M
f, M
f].
So the assertion follows taking B = [−M
Hence for the family of discrete operators defined by
+∞
X
k
(Sw f )(s) =
Γ(ws − k)f
w
k=−∞
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we can apply all the previous theory.
Taking ζw = kΓk1 for every w > 0 and assuming for the sake of simplicity
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that kΓk1 = 1, the class ΥHw (P, (ζw )w>0 ) now takes the form
ΥHw (P, (ζw )w>0 )
(
=
)
Z
∞
X
k
1
≤P
f ∈ Lϕ (R) : lim sup
ϕ λ f
ϕ(λ|f (s)|)ds .
w
w→+∞ w
R
k=−∞
In particular if f ∈ BV (R)∩E ϕ (R), then ϕ◦λ|f | is also of bounded variation,
since ϕ is locally Lipschitz. Thus, using the results in [19], we have
Z
∞
k 1 X
ϕ λ f
lim
=
ϕ(λ|f (s)|)ds,
w→+∞ w
w R
k=−∞
see also [9] and [10]. Now, if f ∈ BV (R) ∩ E ϕ (R), taking F = Cc∞ (R), we
have f − F ⊂ BV (R) ∩ E ϕ (R) and since F ϕ = Lϕ (R), (see [18], [2]), we
obtain the following
Corollary 4.1. Let f ∈ BV (R) ∩ E ϕ (R). Then there exists α > 0 such that
lim ρϕR [α(Sw f − f )] = 0.
w→+∞
If we take ϕ(u) = |u|p , we obtain a convergence result in Lp −spaces.
Thus we obtain a non local modular approximation theorem in Orlicz spaces
for the generalized sampling series of a function f and this improves corresponding results given in [7] and [27] for kernels with compact support.
The above results can be extended also to the nonlinear discrete operators of
the form
+∞
X
k
,
(Tw f )(s) =
K ws − k, f
w
k=−∞
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where the corresponding functions Lw satisfy the previous conditions (see [27],
[23], [6] and [1]).
As a second example, let us consider G = [0, +∞[ with Lebesgue measure
and for every n ∈ N we take Hn = n1 N0 = { nk , k = 0, 1, . . . } endowed with
the measure n1 µc . Let us consider the class of the Szász-Mirak’jan operators
given by
+∞
X
k
(Sn f )(s) =
f
σk,n (s), s ≥ 0,
n
k=0
where
(ns)k
, s ≥ 0.
k!
These operators are generated by the family of functions defined by
k
k
Ln s,
= nL ns, n
= nL(ns, k),
n
n
σk,n (s) = e−ns
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where
sk
.
k!
As shown in [4], the family of kernels Kn : G × Hn × R → R defined by
k
k
Kn s, , u = Ln s,
u
n
n
L(s, k) = e−s
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satisfies all the singularity assumptions and ζn = 1 for every n ∈ N. Now
we show that the family (Ln )n∈N satisfies (∗, ∗). Firstly note that without loss
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J. Ineq. Pure and Appl. Math. 6(4) Art. 123, 2005
of generality we can take C = [0, δ], with δ > 1 a positive integer. Let
B = [0, M ] with M > 1 and log M
> δ. We have
M +1
Z
Ln
G\B
k
s,
n
Z
+∞
ds =
ne−ns
M
(sn)k
ds := I[n, k],
k!
where k ∈ [0, nδ]. By using elementary calculus and a recursion method, we
have
k
X
(nM )j
−nM
I[n, k] = e
j!
j=0
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I[n, k] ≤ e−nM
nδ
X
j=0
(nM )
j!
+∞
nδ −nM X
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j
(nδ)
j!
j=0
n
M nδ −n(M −δ) 1 M δ
=
e
=
.
δ
δ eM −δ
≤
M e
δ
j
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Since M δ < eM −δ , (∗, ∗) follows.
Note that the family of functions (Ln )n∈N also satisfies property (∗) for
every β > 0.
Finally we obtain the following convergence theorem for the Szász-Mirak’jan
operators in Orlicz spaces generated by a convex ϕ-function ϕ.
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+
ϕ
Corollary 4.2. Let f ∈ BV (R+
0 ) ∩ E (R0 ). Then there exists α > 0 such that
lim ρϕR+ [α(Sn f − f )] = 0.
n→+∞
0
Remark 1. Note that, using the characterization in [19], Corollaries 4.1 and
4.2 are still true under the weaker assumption that the function f, (or the function ϕ◦λ|f | ), is locally Riemann integrable and of bounded “coarse” variation
on R+
0 . This notion was introduced in [19] in connection with “simply integrable functions” and it is meaningful if the domain of f is unbounded, since
for bounded domains it is equivalent just to boundedness of the function f.
More generally, let us consider a (increasing) sequence (tn )n∈Z of real numbers such that |tn − tn−1 | ≥ r, for every n ∈ Z and an absolute positive
constant r. Let T = {tn : n ∈ Z}. Let us put G = R and Hw = w1 T . We
provide again G with the Lebesgue measure and we put µw = w1 µc . Let now
Γ : R × T → R be a function such that Γ ∈ L1 (R) with respect to u ∈ R and
P
(Γ.1) +∞
n=−∞ Γ(u, tn ) = 1 for every u ∈ R.
P
(Γ.2) m1 (Γ) := supu∈R +∞
n=−∞ |Γ(u, tn )|{1 + |u − tn |} < +∞.
(Γ.3) For every ε > 0 there exists M > 0 such that
Z
|Γ(s, tn )|ds < ε,
|s|>M
for every tn ∈ T .
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Now, for every w > 0 we define the family of functions Γw : R × Hw →
R on putting
tn
Γw s,
= wΓ(ws, tn ).
w
As before, we can see that the family of kernels Kw : R×Hw ×R → R defined
by
tn
Kw s, , u = wΓ(ws, tn )u
w
satisfies a Lipschitz condition with ψw (u) = u and Lw s, twn = w|Γ(ws, tn )| for
every u ≥ 0, and it is singular.
Note that the property (∗, ∗) is a consequence of (Γ.3) by using the same
arguments as the first example.
Thus the previous theory can be applied to Kramer type operators of the form
X
tn
tn
(Tw f )(s) =
Γw s,
f
w
w
n∈Z
Z
and their nonlinear versions of the form
X
tn
tn
(Tw f )(s) =
Γw s, , f
.
w
w
n∈Z
Finally, some multidimensional versions of discrete operators are included in
our general approach. For example let us consider the multivariate generalized
sampling operator defined by (see [12])
X k
(Sw f )(s) =
f
Γ(ws − k),
w
n
k∈Z
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where s = (s1 , . . . , sn ) ∈ Rn , w = (w1 , . . . , wn ) ∈ Rn+ , k = (k1 , . . . , kn ) ∈
Zn , wk = (k1 /w1 , . . . , kn /wn ), ws = (w1 s1 , . . . , wn sn ) and f : Rn → R. Here
the kernel Γ is given, for example, by a “box” kernel
Γ(s) =
n
Y
Γj (sj ),
j=1
where Γj is a one-dimensional kernel satisfying the above assumptions.
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References
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Properties of Abstract Integral
Operators in Orlicz Spaces and
Applications
Carlo Bardaro and Ilaria Mantellini
[4] C. BARDARO AND I. MANTELLINI, Approximation properties in abstract modular spaces for a class of general sampling type operators, to
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Operators in Orlicz Spaces and
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[16] P.L. BUTZER AND R.L. STENS, Linear prediction by samples from the
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Operators in Orlicz Spaces and
Applications
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On Global Approximation
Properties of Abstract Integral
Operators in Orlicz Spaces and
Applications
Carlo Bardaro and Ilaria Mantellini
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