Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 123, 2005
ON GLOBAL APPROXIMATION PROPERTIES OF ABSTRACT INTEGRAL
OPERATORS IN ORLICZ SPACES AND APPLICATIONS
CARLO BARDARO AND ILARIA MANTELLINI
D EPARTMENT OF M ATHEMATICS AND I NFORMATICS
U NIVERSITY OF P ERUGIA
V IA VANVITELLI 1, 06123 P ERUGIA , ITALY
bardaro@unipg.it
mantell@dipmat.unipg.it
Received 25 August, 2005; accepted 01 September, 2005
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Modular approximation, nonlinear integral operators, regular families, singularity.
2000 Mathematics Subject Classification. 41A25, 41A35, 47G10, 46E30.
1. I NTRODUCTION
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,
Hw
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
253-05
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C ARLO BARDARO AND I LARIA M ANTELLINI
where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of closed subsets
S
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 .
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 wellknown discrete operator, namely the Szász-Mirak’jan operator and we obtain, as an application,
a convergence result in Orlicz space.
2. N OTATIONS AND D EFINITIONS
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 .
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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 :
1
G × H w → R+
0 such that for every w > 0 the sections 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 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
Lw (s, t)dµw (t) ≤ D
0 < βw (s) :=
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.
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
Hw
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
Lw (s, t)dµw (t) = 0.
lim
w→+∞
Hw \Us
3) For every s ∈ G and for every u ∈ R we have
Z
lim
Kw (s, t, u)dµw (t) = u.
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
Z
Lw (s, t)dµw (t) = 0
lim
w→+∞
Hw \Us
uniformly with respect to s ∈ G,
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C ARLO BARDARO AND I LARIA M ANTELLINI
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
Lw (s, t)ψw (|f (t)|)dµw (t) ≤ ψw (kf k∞ )D ≤ M 0 D,
|(Tw f )(s)| ≤
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.
3. C ONVERGENCE
IN
O RLICZ S PACES
+
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 functions.
Moreover we denote by Φ
For ϕ ∈ Φ, we define the functional
Z
ϕ
%G (f ) =
ϕ(|f (s)|)dµG (s)
G
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 subspace of
is the Orlicz space generated by ϕ. If ϕ ∈ Φ,
G
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)
Hw
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 .
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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 ,
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
Lw (s, t)dµG (s) ≤ β
(3.2)
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
e w )w>0 on G = R, such
measure. Let us consider an approximate identity (see [13]) (K
that for every δ > 0
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
e w (s−t)| satisfies propsatisfies the singularity assumptions and the family Lw (s, t) = |K
e w )w>0 of functions K
e w with
erty (∗). For example the above holds for every family (K
supports contained in a fixed interval of the form [−a, a].
2. Let G = Hw = R for every w > 0, provided with Lebesgue measure. Let us consider
the family of functions
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]
Lw (s, t) =
Now it is easy to show that (3.2) holds for every β > 0. This is an example in nonconvolution case (see [4]).
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C ARLO BARDARO AND I LARIA M ANTELLINI
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 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 = (Kw )w>0 ∈ KΞ be singular.
Theorem 3.1. Let ϕ ∈ Φ,
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
%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.
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 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
%G [α(Tw f )χG\B ] ≤
ϕ(αDψw (|f (t)|))dµw (t).
Dwβ Hw
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O N GLOBAL APPROXIMATION PROPERTIES OF ABSTRACT INTEGRAL OPERATORS IN O RLICZ SPACES AND
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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
Letting w → +∞, we have
lim sup %ϕG [α(Tw f )χG\B ]
w→+∞
M
≤ lim sup
ζw
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
%G [α(Tw f )χBk ] =
ϕ(α|(Tw f )(s)|χBk (s))dµG (s)
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 ∈ KΞ be singular. Let
Theorem 3.2. Let ϕ ∈ Φ,
β
us assume that w 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→+∞
for every f ∈ F.
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 the triple (%ϕ , ψw , %η ) is
Theorem 3.3. Let ϕ ∈ Φ,
Hw
Hw
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
P
lim sup %ϕG [α(Tw f − Tw g)] ≤ %ηG [λ(f − g)].
D
w→+∞
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C ARLO BARDARO AND I LARIA M ANTELLINI
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
Z
ϕ
%G [α(Tw f − Tw g)] =
ϕ(α|(Tw f )(s) − (Tw g)(s)|)dµG (s)
G
Z
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
ζw
cw 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.
Now we are ready to prove the main theorem of this section.
e η ∈ Φ and Ξ = (ψw )w>0 ⊂ Ψ be such that the triple (%ϕ , ψw , %η ) is
Theorem 3.4. Let ϕ ∈ Φ,
Hw
Hw
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.
w→+∞
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
w→+∞
and by Theorem 3.3 we have
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
P
P
P
lim sup %ϕG [α(Tw f − f )] ≤ %ηG [λ0 (f − fn )] + %ϕG [λ0 (f − fn )] < ε .
D
D
D
w→+∞
The assertion follows since ε is arbitrary.
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Now we describe a different approach to the convergence problem by using a more general assumption than property (∗). This assumption has been used in [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 ∈ KΞ be singular. Let f ∈
Theorem 3.5. Let ϕ ∈ Φ,
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
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.
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
ε
εR
≤
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 the triple (%ϕ , ψw , %η ) is
Theorem 3.6. Let ϕ ∈ Φ,
Hw
Hw
properly directed. Let K = (Kw )w>0 ∈ KΞ be singular. Let us assume that the family
(Lw (·, t))t∈Hw ,w>0 satisfies property (∗∗) and the family of measures (µw )w>0 is locally bounded.
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C ARLO BARDARO AND I LARIA M ANTELLINI
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→+∞
4. A PPLICATIONS TO D ISCRETE O PERATORS
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) +∞
= 1 for every u ∈ R.
k=−∞ Γ(u − k) P
(L.2) m0 (Γ) := supu∈R +∞
k=−∞ |Γ(u − k)| < +∞.
Note that from the summability of Γ we deduce the following property
(L.3) For every ε > 0 there exists M > 0 such that
Z
|Γ(s)|ds < ε.
|s|>M
Now, for every w > 0, we define
k
k
Γw s,
= wΓ w s −
= wΓ(ws − k), s ∈ R, k ∈ Z.
w
w
We prove that the family of kernels Kw : G × Hw × R → R, defined by
k
Kw s, , u = wΓ(ws − k)u,
w
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
|ws−k|>δw
By Lemma 1 in [26], assumption (L.2) implies
X
lim
|Γ(u − k)| = 0,
R→+∞
|u−k|>R
uniformly with respect to u ∈ R and so for a fixed ε > 0 there is Rε > 0 such that
X
|Γ(u − k)| < ε,
|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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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 ε > 0 be fixed. Let M be the constant in (L.3) and let us
f > 0 such that M
f − δ > M. Since k ∈ [−δ, δ], we only consider k ∈ [−δw, δw]. So
choose M
w
we have, with w > 1,
Z
Z −M
Z +∞
fw−k
w|Γ(ws − k)|ds =
|Γ(u)|du +
|Γ(u)|du
f
fw−k
|s|>M
−∞
M
Z
≤
|Γ(u)|du
f−δ)w
|u|>(M
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=−∞
we can apply all the previous theory.
Taking ζw = kΓk1 for every w > 0 and assuming for the sake of simplicity that kΓk1 = 1, the
class ΥHw (P, (ζw )w>0 ) now takes the form
ΥHw (P, (ζw )w>0 )
(
=
)
Z
∞
k
1 X
≤P
f ∈ L (R) : lim sup
ϕ λ f
ϕ(λ|f (s)|)ds .
w
w→+∞ w
R
k=−∞
ϕ
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
∞
1 X
k lim
ϕ λ f
=
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=−∞
where the corresponding functions Lw satisfy the previous conditions (see [27], [23], [6] and
[1]).
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C ARLO BARDARO AND I LARIA M ANTELLINI
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
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
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 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
Z +∞
k
(sn)k
Ln s,
ds =
ds := I[n, k],
ne−ns
n
k!
G\B
M
where k ∈ [0, nδ]. By using elementary calculus and a recursion method, we have
I[n, k] = e−nM
k
X
(nM )j
j=0
j!
and so
−nM
I[n, k] ≤ e
nδ
X
(nM )j
j=0
j!
+∞
M nδ e−nM X (nδ)j
δ
j!
j=0
n
M nδ −n(M −δ) 1 M δ
=
e
=
.
δ
δ eM −δ
≤
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.
+
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→+∞
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Remark 4.3. 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) +∞
1 for every u ∈ R.
n=−∞ Γ(u, tn ) =
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 .
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
(Tw f )(s) =
X
Γw
n∈Z
tn
s, , f
w
tn
w
.
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
where s = (s1 , . . . , sn ) ∈ Rn , w = (w1 , . . . , wn ) ∈ Rn+ , k = (k1 , . . . , kn ) ∈ Zn , kw =
(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
n
Y
Γ(s) =
Γj (sj ),
j=1
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C ARLO BARDARO AND I LARIA M ANTELLINI
where Γj is a one-dimensional kernel satisfying the above assumptions.
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