Danilo Costarelli, Gianluca Vinti
Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces
To Professor Julian Musielak, a Master, a Mentor and a sincere friend, with high esteem and friendship
Abstract. In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively.
The general setting of Orlicz spaces allows to deduce directly the results concerning the rate of convergence in Lp-spaces, 1 ≤ p < ∞, very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fejér and B-spline kernels are also considered.
1991 Mathematics Subject Classification: 41A25, 41A30, 46E30, 47A58, 47B38, 94A12.
Key words and phrases: Nonlinear sampling Kantorovich operators, Orlicz spaces, order of approximation, Lipschitz classes, irregular sampling.
1. Introduction. In this paper, we study the rate of convergence for the nonlinear sampling Kantorovich operators introduced in [57] and we generalize the results obtained in [31] to the nonlinear setting. Here, we consider both the case of uniform approximation, for uniformly continuous and bounded functions belonging to Lipschitz classes and the modular approximation for functions belonging to Orlicz spaces. In the latter case, we introduce Lipschitz class of the Zygmund-type which take into account of the modular functional involved (see e.g. [31]).
The theory of nonlinear integral operators in connection with approximation problems has been started in [42, 44, 45, 46] by the work of the polish mathematician J. Musielak, a direct student of W. Orlicz. Successively, in [10] the above theory
has been carried out in collaboration with C. Bardaro and G. Vinti in the general context of modular spaces, such as in Orlicz spaces (see e.g. [6, 7, 12, 8, 9, 15, 14, 47, 48, 16, 55, 41]).
The sampling operators of the Kantorovich-type were first introduced in [3] by Bardaro, Butzer, Stens and Vinti, in the linear case and in one dimensional setting.
Later on, various extensions of the theory have been produced; for instance, in [29, 30] the multivariate theory of the sampling Kantorovich operators has been studied in the linear and nonlinear setting, with applications to Image Processing.
Moreover, extensions to a more general context are given in [58, 5].
Nonlinear sampling Kantorovich operators are important both from the mathe- matical and the applications point of view; as a matter of fact there are signals that need to be reconstructed by using a nonlinear process.
The nonlinear sampling Kantorovich operators here considered (introduced in Section ), are of the form
(Swf )(x) := X
k∈Z
χ wx− tk, w
∆k
Z tk+1/w tk/w
f (u)du
!
, x∈ R, (I)
where f : R → R is a locally integrable function such that the series is convergent for every x ∈ R, χ : R2→ R represents a kernel function satisfying certain properties, and (tk)k∈Z is a suitable strictly increasing sequence of real numbers with ∆k :=
tk+1− tk > 0, k ∈ Z. Examples of kernels χ satisfying the assumptions of the theory of the present paper, can be constructed using, e.g., central B-splines of order n ∈ N+, or the Fejér’s kernel (see e.g. [22, 10, 3, 57]). The choice of (tk)k∈Z
allows us to sample signals by an irregular sampling scheme. If tk = k, k ∈ Z, we obtain the uniform spaced sampling.
The main results of the paper show that, for f belonging to a Lipschitz class and under certain assumptions,
kSwf− fk∞ = O(w−ε), as w→ +∞, for suitable ε > 0 (see Theorem 4.1 of Section ) and
Iϕ[µ(Swf − f)] = O(w−θ), as w→ +∞,
for suitable constants µ, θ > 0, where Iϕ denotes the modular of the Orlicz space under consideration (see Theorem 5.2 of Section ).
The importance of the above operators (I) lies in theory of Signal Processing, where they can be used to reconstruct and approximate signals; in a multivariate context the above operators are suitable to reconstruct and approximate images (see [29, 30, 28]).
The operators (I), represent the natural extension of the nonlinear generalized sampling operators to the Lp-setting, 1 ≤ p < +∞. These latter operators, were first introduced by the German mathematician P.L. Butzer and his school at Aachen in the linear form and subsequently, they were extended to the nonlinear setting in [14]. For references on sampling series see e.g. [19, 23, 52, 20, 24, 25, 21, 26, 27, 11, 13, 14, 54, 55, 17, 1, 33, 4]. The theory of the generalized sampling operators
represents an approximate version of the classical Sampling Theory, based on the classical Whittaker-Kotelnikov-Shannon sampling theorem (see e.g. [53, 37, 32, 34, 18, 35, 36, 2]).
One of the advantages, with respect to the Butzer’s operators, given by the series of the form (I) is that, instead of the sampling values f(k/w), one has an average of f on a small interval containing k/w (here instead of k we have tk). In practice, this situation very often occurs in Signal Processing when one cannot match exactly the node tk: this represents the so-called time-jitter error. Therefore, the sampling Kantorovich operators reduces time-jitter errors calculating the information in a neighborhood of a point rather than exactly at that point. The integrals in (I) and in particular the presence of the mean values, become our operators suitable for the study in Lp-spaces, or more generally in Orlicz spaces Lϕ(R).
We point out that, the setting of Orlicz spaces is important since allows the reconstruction of not necessarily continuous signals; this situation very often oc- curs especially in the applications to Image Processing, where digital images are represented by discontinuous functions.
Important examples of Orlicz spaces here considered, are given by the Zygmund spaces, also known as interpolation spaces, very important e.g. in the theory of partial differential equations, and the exponential spaces, very used for embedding theorems between Sobolev spaces, see e.g. [43, 10, 3].
Finally, in Section we furnish applications to concrete operators constructed by means of special kernels, as e.g. Fejér or B-spline kernels.
2. Preliminaries. In this section, we introduce some notations that will be very useful along the paper. In what follows, we denote by C(R) the set of all uniformly continuous and bounded functions f : R → R endowed with the usual sup-norm k · k∞.
In order to study the rate of convergence of a family of nonlinear discrete ope- rators in C(R), we introduce the Lipschitz class of the Zygmund-type with which we will work. We define Lip∞(ν), 0 < ν ≤ 1, by
Lip∞(ν) := {f ∈ C(R) : kf(·) − f(· + t)k∞=O(|t|ν), as t → 0} , where for any two functions f, g : R → R, f(t) = O(g(t)) as t → 0, means that there exist constants C, γ > 0 such that |f(t)| ≤ C |g(t)| for every |t| ≤ γ ( see e.g.
[10, 56]).
We now recall the definition and some basic properties of Orlicz spaces.
We will say that a function ϕ : R+0 → R+0 is a ϕ-function, if it satisfies the following conditions:
1. ϕ is a continuous function with ϕ(0) = 0;
2. ϕ is a non-decreasing function and ϕ(u) > 0 for every u > 0;
3. lim
u→+∞ϕ(u) = +∞.
Let now introduce the functional Iϕ associated to the ϕ-function ϕ and defined by Iϕ[f ] :=
Z
Rϕ(|f(x)|) dx,
for every f ∈ M(R), i.e. for every (Lebesgue) measurable function f : R → R. As it is well-known, Iϕ is a modular functional (see e.g. [43, 50, 10]). It is easy to prove that if ϕ is convex than Iϕis convex.
The Orlicz space generated by ϕ is defined by
Lϕ(R) := {f ∈ M(R) : Iϕ[λf ] < +∞, for some λ > 0} .
A natural notion of convergence in Orlicz spaces, called modular convergence, can be introduced (see [49]).
Namely, we will say that a net of functions (fw)w>0 ⊂ Lϕ(R) is modularly convergent to f ∈ Lϕ(R), if there exists λ > 0 such that
Iϕ[λ(fw− f)] = Z
Rϕ(λ|fw(x)− f(x)|) dx → 0, as w→ +∞.
Moreover, we recall, for the sake of completeness, that in Lϕ(R) can be also given a strong notion of convergence, i.e., the Luxemburg-norm convergence, see e.g. [43, 10]. The modular convergence induces a topology in Lϕ(R) called modular topology.
Now, we introduce the Lipschitz class used in Orlicz spaces in order to study the rate of convergence for the nonlinear sampling Kantorovich operators we will deal with.
By Lipϕ(ν), 0 < ν ≤ 1, we define the set of all functions f ∈ M(R) such that, there exists λ > 0 with
Iϕ[λ(f (·) − f(· + t))] = Z
Rϕ (λ|f (x) − f (x + t)|) dx = O(|t|ν), as t → 0, see [31].
For further results concerning Orlicz spaces, see [39, 43, 38, 40, 50, 51].
3. Nonlinear sampling Kantorovich operators. In this section, we first recall the definition of the nonlinear sampling Kantorovich operators introduced in [57].
Let π = (tk)k∈Zbe a sequence of real numbers such that −∞ < tk< tk+1< +∞ for every k ∈ Z, limk→±∞ tk = ±∞ and such that δ ≤ ∆k := tk+1− tk ≤ ∆, for every k ∈ Z and for some δ, ∆ > 0.
In what follows, a function χ : R × R → R will be called a kernel if it satisfies the following conditions:
(χ1) k7→ χ(wx − tk, u)∈ `1(Z), for every (x, u) ∈ R2 and w > 0;
(χ2) χ(x, 0) = 0, for every x ∈ R;
(χ3) χis a (L, ψ)-Lipschitz kernel, i.e., there exist a measurable function L : R → R+0 and a ϕ-function ψ, such that
|χ(x, u) − χ(x, v)| ≤ L(x)ψ(|u − v|), for every x, u, v ∈ R;
(χ4) there exists θ0> 0such that
Tw(x) := sup
u6=0
1 u
X
k∈Z
χ(wx− tk, u)− 1
= O(w−θ0) as w → +∞, uniformly with respect to x ∈ R.
Moreover, we will assume that the function L of condition (χ3) satisfies the following properties:
(L1) L∈ L1(R) and is bounded in a neighborhood of the origin;
(L2) there exists β0> 0such that mβ0,π(L) := sup
x∈R
X
k∈Z
L(wx− tk)|wx − tk|β0 < +∞;
(L3) there exists α0> 0such that, for every M > 0, w
Z
|x|>M
L(wx) dx = O(w−α0), as w→ +∞.
Remark 3.1 (a) Note that, if L(x) = O(x−1−β0−ε) as x → ±∞, for some ε > 0, condition (L2) holds for β0, see [22, 3, 31].
(b) If the function L has compact support, i.e., supp L ⊂ [−B, B], B > 0, condition (L2) is satisfied for every β0> 0.
(c) If supp L ⊂ [−B, B], B > 0, we obtain, for every M > 0,
w Z
|x|>M
L(wx) dx = Z
|u|>wM
L(u) du = 0,
for sufficiently large w > B/M; then condition (L3) is fulfilled for every α0> 0.
(d) Condition (L3) is used only to study the nonlinear sampling Kantorovich ope- rators in the setting of Orlicz spaces.
We now recall the definition of the nonlinear sampling Kantorovich operators for a given kernel χ. We define
(Swf )(x) := X
k∈Z
χ wx− tk, w
∆k
Z tk+1/w tk/w
f (u)du
!
, x∈ R,
where f : R → R is a locally integrable function such that the series is convergent for every x ∈ R.
From now on, we will consider functions on the domain of the operators (Sw)w>0.
Remark 3.2 Note that, if the kernel χ is of the form χ(x, u) = L(x) u, with L satisfying conditions (L1), (L2) and (L3), our operators reduces to the linear ones introduced in [3]. In this case condition (χ4) becomes
(1) Tw(x) =
X
k∈Z
L(wx− tk)− 1
= O(w−θ0), as w→ +∞,
uniformly with respect to x ∈ R, for some θ0> 0. Sometimes, in place of condition (1) we require a stronger condition, i.e.,
(2) X
k∈Z
L(u− tk) = 1,
for every u ∈ R. In this case condition (χ4) holds for every θ0 > 0. When tk = k (uniform sampling scheme), (2) is equivalent to
L(k) :=b
0, k∈ Z \ {0} , 1, k = 0, where bL(v) :=
Z
RL(u)e−ivu du, v ∈ R, is the Fourier transform of L; see [22, 25, 3, 29, 31].
We now recall the following lemma, that will be very useful in the paper.
Lemma 3.3 ([31]) Let L be a function satisfying conditions (L1) and (L2). Then, (i) m0,π(L) := supx∈RP
k∈ZL(wx− tk) < +∞;
(ii) for every γ > 0 X
|wx−tk|>γw
L(wx− tk) = O(w−β0), as w→ +∞,
uniformly with respect to x ∈ R, where β0> 0is the constant of condition (L2).
Remark 3.4 Note that, (i) of Lemma 3.3, together with assumptions (χ2) and (χ3) ensures that the functionals (Sw(·))w>0 are well-defined for functions f ∈ L∞(R).
Indeed,
|(Swf )(x)| ≤ X
k∈Z
L(wx− tk) ψ(kfk∞) ≤ m0,π(L) ψ(kfk∞) < +∞,
for every x ∈ R, i.e., Sw: L∞(R) → L∞(R).
4. Order of approximation in C(R). We will now study the order of approximation of the family of nonlinear sampling Kantorovich operators in C(R).
We have the following.
Theorem 4.1 Let χ be a kernel and suppose in addition that ψ of condition (χ3) satisfies the following assumption:
(3) ψ(u) = O(up), as u→ 0+,
for some 0 < p ≤ 1.
Then, for every f ∈ Lip∞(ν), 0 < ν ≤ 1, we have:
kSwf− fk∞ = O(w−), as w→ +∞,
with := min {νp, β0, θ0}, where β0and θ0 are the positive constants of conditions (L2) and (χ4), respectively.
Proof We first consider the case of kernel χ satisfying condition (χ3) with L such that condition (L2) is satisfied for 0 < β0≤ p.
By condition (3) there exists M > 0 and 0 < u < 1 such that ψ(u) ≤ Mup, for every u ∈ R+0 with u ≤ u. Let now f ∈ Lip∞(ν), with 0 < ν ≤ β0/p, be fixed. We know that there are C1> 0and γ > 0 such that
sup
x∈R|f(x) − f(x + t)| ≤ C1 |t|ν,
for every |t| ≤ γ. Without loss of generality, we can assume γ > 0 such that (γ/2)ν≤ u/2. Now, let x ∈ R be fixed. We can write
|(Swf )(x)− f(x)|
≤ |(Swf )(x)−X
k∈Z
χ (wx− tk, f (x))| + |X
k∈Z
χ (wx− tk, f (x))− f(x)|
≤X
k∈Z
L (wx− tk) ψ
w
∆k
Z tk+1/w tk/w
f (u)du− f(x)
! +
X
k∈Z
χ (wx− tk, f (x))− f(x)
≤
X
|wx−tk|≤wγ/2
+ X
|wx−tk|>wγ/2
L (wx− tk) ψ w
∆k
Z tk+1/w
tk/w |f(u) − f(x)| du
!
+
X
k∈Z
χ (wx− tk, f (x))− f(x)
=: I1 + I2 + I3,
for every w > 0. By the change of variable u = x + t in the integrals of I1, we have,
I1 = X
|wx−tk|≤wγ/2
L (wx− tk) ψ w
∆k
Z (tk+1/w)−x
(tk/w)−x |f(x + t) − f(x)| dt
! .
Now, for every t ∈ [tk/w− x, tk+1/w− x], if |wx − tk| ≤ wγ/2, we have
|t| ≤ |t − tk/w + x| + |tk/w− x| ≤ ∆ w+γ
2 < γ,
for sufficiently large w > 0, then, since f ∈ Lip∞(ν), we obtain
(4) I1 ≤ X
|wx−tk|≤wγ/2
L (wx− tk) ψ C1 w
∆k
Z (tk+1/w)−x (tk/w)−x |t|νdt
! .
In order to estimate (4), we introduce the following notations.
Denote by L = {k ∈ Z : |wx − tk| ≤ wγ/2}, L1 = {k ∈ L : tk/w≥ x} and L2 = {k ∈ L : tk+1/w≤ x}, with w > 0. In general L = L1∪ L2∪ L3, where L3 ={k} if there exists k ∈ Z such that tk/w < x < tk+1/w, otherwise L3 =∅.
In what follows we consider only the case of L3={k} (the case L3=∅ is similar).
Now, we rewrite the right-hand side of (4) as follows X
|wx−tk|≤wγ/2
L(wx− tk) ψ C1
w
∆k
Z (tk+1/w)−x (tk/w)−x |t|ν dt
!
= X
k∈L1
+ X
k∈L2
+ X
k∈L3
!
L(wx− tk) ψ C1 w
∆k
Z (tk+1/w)−x (tk/w)−x |t|ν dt
!
=: J1+ J2+ J3. We first estimate J1. We obtain
J1≤ X
k∈L1
L(wx−tk) ψ
C1 sup
t∈[tkw−x,tk+1w −x]
|t|ν
≤ X
k∈L1
L(wx−tk) ψ (C1 |tk+1/w− x|ν) .
Now, since k ∈ L1 ⊂ L and | · |ν, 0 < ν ≤ 1, is concave and then subadditive, we have
|tk+1/w− x|ν =
tk+ ∆k− wx w
ν
≤
tk− wx w
ν
+
∆ w
ν
≤ γ 2
ν +
∆ w
ν
≤ u 2 +
∆ w
ν
≤ u for sufficiently large w > 0; then by condition (3)
J1 ≤ M C1p wνp
X
k∈L1
L(wx− tk)|tk+ ∆k− wx|νp, (5)
for sufficiently large w > 0. Observing again that νp ≤ β0≤ p ≤ 1, | · |νp is concave and then subadditive; hence it turns out that
J1 ≤ M C1p wνp
X
k∈L1
L(wx− tk) [|tk− wx|νp + ∆νpk ]
≤ M C1p wνp
X
k∈L1
L(wx− tk) |wx − tk|νp + M C1p∆νp wνp
X
k∈L1
L(wx− tk).
We now estimate J2. Since k ∈ L2⊂ L, it follows that |tk/w− x|ν ≤ (γ/2)ν ≤ u.
Then, by condition (3) we obtain
J2 ≤ X
k∈L2
L(wx− tk) ψ
C1 sup
t∈[tkw−x,tk+1w −x]
|t|ν
≤ X
k∈L2
L(wx− tk) ψ (C1|tk/w− x|ν) ≤ M C1p wνp
X
k∈L2
L(wx− tk)|wx − tk|νp, for sufficiently large w > 0. Finally, we estimate J3. Using again the subadditivity of | · |νpand condition (3), we obtain
J3 ≤ L(wx − tk) ψ
C1 sup
t∈htk
w−x,tk+1w −xi|t|ν
≤ L(wx − tk) ψ C1max tk/w− x ν, tk+1/w− x ν
≤ M C1p
wνp L(wx− tk) tk− wx νp + ∆νp
= M C1p
wνp L(wx− tk) wx − tk
νp + M C1p∆νp
wνp L(wx− tk), for sufficiently large w > 0. Then, by the above estimates we can state that
J1+ J2+ J3 ≤ M C1p wνp
(X
k∈L
L(wx− tk)|wx − tk|νp + ∆νp X
k∈L1∪L3
L(wx− tk) )
(6) ≤ M C1p
wνp {mνp,π(L) + ∆νpm0,π(L)} ,
for every x ∈ R and for sufficiently large w > 0. Now, since if mβ0,π(L) < +∞ then mα,π(L) < +∞ for every 0 < α ≤ β0 and since νp ≤ β0, we obtain from (6) that:
J1+ J2+ J3 ≤ M C1p
wνp {mνp,π(L) + ∆νpm0,π(L)} =: eC w−νp < +∞, for every sufficiently large w > 0; then I1=O(w−νp), as w → +∞.
We now estimate I2. We have, as a consequence of Lemma 3.3 (ii), I2 ≤ ψ (2kfk∞) X
|wx−tk|>wγ/2
L (wx− tk) = O(w−β0), as w→ +∞.
Finally, we estimate I3. Obviously, since χ(x, 0) = 0, if f(x) = 0 it turns out that I3= 0. While, for f(x) 6= 0 we have
I3 =
X
k∈Z
χ(wx− tk, f (x))− f(x)
≤ |f(x)| Tw(x);
then we obtain by condition (χ4)
I3 ≤ kfk∞Tw(x) =O(w−θ0), as w→ +∞, uniformly with respect to x ∈ R. In conclusion, we have
kSwf− fk∞ = O(w−), as w→ +∞, with := min {νp, β0, θ0}.
Let now consider the case f ∈ Lip∞(ν)with β0/p≤ ν ≤ 1 be fixed. It is easy to observe that, since Lip∞(ν)⊆ Lip∞(β0/p), we can immediately reduces to the above case, from which we obtain that
kSwf− fk∞ = O(w−), as w→ +∞, with := min {β0, θ0} = min {νp, β0, θ0}.
Finally, we study the case of χ satisfying condition (χ3) with L such that, condition (L2) is satisfied for β0 > p. As noted above, mβ0,π(L) < +∞ implies mp,π(L) < +∞, then L satisfies condition (L2) also for β0 = p and thus we can reduces to the previous case. This completes the proof.
5. Order of approximation in Orlicz spaces. We will now study the order of approximation for the nonlinear sampling Kantorovich operators in the general setting of Orlicz spaces. In order to obtain results in this direction, we will need of a growth condition on the composition of the function ϕ, which generates the Orlicz space and the function ψ of condition (χ3).
Namely, given a ϕ-function ϕ we require that there exists a ϕ-function η such that, for every λ ∈ (0, 1), there exists a constant Cλ∈ (0, 1) satisfying
(H) ϕ(Cλψ(u)) ≤ η(λu), u∈ R+0,
where ψ is the ϕ-function of condition (χ3) (see [10, 57, 30]).
In what follows, we will write ϕ ∈ Hη to denote that the above condition (H) is satisfied for ϕ and η convex.
First of all, we recall a modular continuity property for our operators. For a proof see [57].
Theorem 5.1 ([57]) Let ϕ ∈ Hη and f, g ∈ Lη(R). Then, for every λ ∈ (0, 1), there exists µ > 0 such that
Iϕ[µ(Swf− Swg)] ≤ 1
δm0,π(L)kLk1Iη[λ(f− g)].
In particular, for g ≡ 0, we have that
Sw: Lη(R) → Lϕ(R).
Now, we are ready to investigate the rate of convergence for the family of our nonlinear operators in Orlicz spaces.
Theorem 5.2 Let χ be a kernel, ϕ ∈ Hη and f ∈ Lϕ+η(R) ∩ Lipη(ν), 0 < ν ≤ 1.
Suppose in addition that, there exist α1, γ > 0 such that
(7) w
Z
|t|≤γ
L(wt)|t|νdt = O(w−α1), as w→ +∞,
where L is the function of condition (χ3). Then, there exists µ > 0 such that Iϕ[µ(Swf− f)] = O(w−θ), as w→ +∞,
with θ := min {ν, α0, α1, θ0}, where α0, θ0 > 0 are the constants of conditions (L3) and (χ4), respectively.
Proof First of all, since f ∈ Lϕ+η(R) ∩ Lipη(ν), we have that there exist λ1, λ2> 0such that Iϕ+η[λ1f ] < +∞ and
Iη[λ2(f (·) − f(· + t))] = O(|t|ν), as t→ 0, i.e., there exist M1, γ > 0 such that
Iη[λ2(f (·) − f(· + t))] ≤ M1|t|ν, for every |t| ≤ γ. Now, we fix λ > 0 such that
λ < min{1, λ1/2, λ2} .
In correspondence to λ, by condition (H) there exists Cλ∈ (0, 1) such that ϕ(Cλψ(u))≤ η(λu), u ∈ R+0, while by condition (χ4) there exists M2> 0such that
Tw(x)≤ M2w−θ0,
uniformly with respect to x ∈ R, for sufficiently large w > 0. Now we choose µ > 0 such that
µ < min{Cλ/(3m0,π(L)), λ1/(3M2)} . By the properties of Iϕwe can write:
Iϕ[µ(Swf− f)] = Z
Rϕ(µ|(Swf )(x)− f(x)|) dx
≤1 3
(Z
Rϕ 3µ
(Swf )(x)−X
k∈Z
χ wx− tk, w
∆k
Z tk+1/w tk/w
f (u + x−tk
w)du!
! dx
+ Z
Rϕ 3µ
X
k∈Z
χ wx− tk, w
∆k
Z tk+1/w tk/w
f (u + x−tk
w)du
!
−X
k∈Z
χ (wx− tk, f (x))
! dx
+ Z
Rϕ 3µ
X
k∈Z
χ (wx− tk, f (x))− f(x)
! dx
)
=: 1
3{J1 + J2 + J3} .
First, we estimate J1. By condition (χ3), and applying Jensen’s inequality and Fubini-Tonelli theorem, we obtain
J1= Z
R
ϕ 3µ
(Swf )(x)−X
k∈Z
χ wx− tk, w
∆k
Z tk+1/w tk/w
f (u + x−tk
w)du!
! dx
≤ Z
R
ϕ 3µX
k∈Z
L (wx− tk) ψ w
∆k
Ztk+1/w
tk/w |f(u) − f(u + x −tk
w)|du
!!
dx
≤ 1
m0,π(L) Z
R
X
k∈Z
L(wx− tk)ϕ 3µm0,π(L) ψ w
∆k
Z tk+1/w
tk/w |f(u) − f(u + x −tk
w)|du
!!
dx
(8)
= 1
m0,π(L) X
k∈Z
Z
RL(wx− tk)ϕ 3µm0,π(L)ψ w
∆k
Z tk+1/w
tk/w |f(u) − f(u + x −tk
w)|du
!!
dx.
Putting t = x−tk/win (8), using condition (H), and applying again Fubini-Tonelli theorem and Jensen’s inequality, we may write
J1≤ 1 m0,π(L)
X
k∈Z
Z
RL(wt)ϕ 3µm0,π(L)ψ w
∆k
Z tk+1/w
tk/w |f(u) − f(u + t)| du
!!
dt
≤ 1
m0,π(L) X
k∈Z
Z
RL(wt) ϕ Cλψ w
∆k
Z tk+1/w
tk/w |f(u) − f(u + t)| du
!!
dt
≤ 1
m0,π(L) X
k∈Z
Z
RL(wt) η λw
∆k
Z tk+1/w
tk/w |f(u) − f(u + t)|du
! dt
≤ 1
m0,π(L) X
k∈Z
Z
RL(wt) w
∆k
Z tk+1/w tk/w
η (λ|f(u) − f(u + t)|) dudt
≤ 1
δm0,π(L) Z
Rw L(wt) X
k∈Z
Z tk+1/w tk/w
η (λ|f(u) − f(u + t)|) dudt
≤ 1
δm0,π(L) Z
Rw L(wt)
Z
Rη (λ|f(u) − f(u + t)|) du
dt
= 1
δm0,π(L) (Z
|t|≤eγ
w L(wt)
Z
Rη (λ|f(u) − f(u + t)|) du
dt +
+ Z
|t|>eγ
w L(wt)
Z
Rη (λ|f(u) − f(u + t)|) du
dt
)
=: 1
δm0,π(L){J1,1+ J1,2} ,
with eγ := min {γ, γ}, where γ > 0 is the constant of condition (7). Now, since f ∈ Lipη(ν), and using (7), we obtain that:
J1,1 ≤ M1 Z
|t|≤eγ
w L(wt)|t|νdt = O(w−α1), as w→ +∞.
For J1,2, by the convexity of η, we have
J1,2 ≤ Z
|t|>eγ
wL(wt)1 2
Z
Rη(2λ|f(u)|)du + Z
Rη(2λ|f(u + t)|)du
dt.
Noting that Z
Rη(2λ|f(u + t)|) du = Z
Rη(2λ|f(u)|) du, for every t ∈ R, we obtain
J1,2 ≤ Z
|t|>eγ
w L(wt)
Z
Rη(2λ|f(u)|)du
dt ≤ Iη[λ1f ] w Z
|t|>eγ
L(wt)dt < +∞, for w > 0; hence J1,2=O(w−α0), as w → +∞, by condition (L3) with α0> 0.
Now, we estimate J2. By condition (χ3), putting t = u−tk/wand using Jensen’s inequality, we have
J2≤ Z
Rϕ 3µX
k∈Z
L (wx− tk) ψ w
∆k
Z tk+1/w tk/w
f (u + x−tk
w)du− f(x)
!!
dx
= Z
Rϕ 3µX
k∈Z
L (wx− tk) ψ
w
∆k
Z ∆k/w 0
f (t + x) dt − f(x)
!!
dx
≤ Z
Rϕ 3µX
k∈Z
L (wx− tk) ψ w
∆k
Z ∆k/w
0 |f(t + x) − f(x)| dt
!!
dx
≤ 1
m0,π(L) Z
R
X
k∈Z
L (wx− tk) ϕ 3µm0,π(L)ψ w
∆k
Z ∆k/w
0 |f(t + x) − f(x)|dt
!!
dx
≤ 1
m0,π(L) Z
R
X
k∈Z
L (wx− tk) ϕ Cλψ w
∆k
Z ∆k/w
0 |f(t + x) − f(x)|dt
!!
dx
≤ 1
m0,π(L) Z
R
X
k∈Z
L (wx− tk) η λ w
∆k
Z ∆k/w
0 |f(t + x) − f(x)| dt
! dx.
Applying again Jensen’s inequality and Fubini-Tonelli theorem and since f ∈ Lipη(ν), we have
J2 ≤ 1
m0,π(L) Z
R
X
k∈Z
L (wx− tk)
"
w
∆k
Z ∆k/w 0
η (λ|f(t + x) − f(x)|) dt
# dx.
≤ 1 δ m0,π(L)
Z
RwX
k∈Z
L (wx− tk)
"Z ∆/w 0
η (λ|f(t + x) − f(x)|) dt
# dx.
≤ 1
δ m0,π(L) Z
Rm0,π(L) w
"Z ∆/w 0
η (λ|f(t + x) − f(x)|) dt
# dx.
= w δ
Z ∆/w 0
Z
Rη (λ|f(t + x) − f(x)|) dx
dt≤ M1
δ w Z ∆/w
0 |t|ν dt, for sufficiently large w > 0. Now, we obtain
w Z ∆/w
0 |t|ν dt = Z ∆
0 |u
w|ν du = w−ν Z ∆
0 |u|ν du =O(w−ν), as w → +∞, then,
J2 = O(w−ν), as w→ +∞.
Finally, denoted by A0⊆ R the set of all points of R for which f 6= 0 a. e., we have
J3 = Z
A0
ϕ 3µ
X
k∈Z
χ (wx− tk, f (x))− f(x)
! dx ≤
Z
A0
ϕ (3µ|f(x)| Tw(x)) dx.
Now, using the convexity of ϕ and condition (χ4), we obtain
J3 ≤ Z
A0
ϕ (3µ|f(x)| Tw(x)) dx ≤ Z
A0
ϕ 3M2µ w−θ0|f(x)| dx
≤ w−θ0 Z
A0
ϕ (3M2µ |f(x)|) dx ≤ w−θ0Iϕ[λ1f ] < +∞,
for every sufficiently large w > 0, i.e., J3 =O(w−θ0), as w → +∞. In conclusion, we have
Iϕ[µ(Swf− f)] = O(w−θ), as w→ +∞,
where θ := min {ν, α0, α1, θ0}.
Note that, under suitable assumptions on the kernels, condition (7) is satisfied. For instance, in the special case of kernels χ satisfying condition (χ3) with L having compact support, e.g. supp L ⊂ [−B, B], B > 0, we have that
(9) Z
|t|≤γ
w L(wt)|t|νdt≤ Z
|u|≤B
L(u) u w
ν du =: Kw−ν,
for sufficiently large w > 0, i.e., α1= ν. Moreover, as noted in Remark 3.1 (c), in the above case L satisfies condition (L3) for every α0 > 0. Hence, we obtain the following.
