On the convergence properties of Durrmeyer-Sampling Type Operators in Orlicz spaces

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces $L^\varphi(\mathbb{R})$. From the latter result, the convergence in $L^p(\mathbb{R})$-space, $L^\alpha\log^\beta L$, and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.


Introduction
The theory of sampling series, in one and several variables, is one of the most studied topics in the approximation theory, in view of its many applications, especially in signal and image processing. Sampling-type operators have been introduced in order to study approximate version of the well-known Wittaker-Kotel'nikov-Shannon sampling theorem (see, e.g., [43,10,15,30,31,32]). Among the most studied families of sampling operators, we can find the celebrated family of the generalized (see, e.g., [40,12,11,14,44,45,1]) and Kantorovich type series (see, e.g., [4,37]), that have been introduced in the 80s and in 2007, respectively, thanks to the crucial contribution of the German mathematician P.L. Butzer and his coauthors. The aim of this work is to extend the main approximation properties, including convergence results and quantitative estimates, for the so-called Durrmeyer sampling type series, introduced by C. Bardaro and I. Mantellini in [7]. Durrmeyer sampling operators represent a generalization of the generalized and of the Kantorovich sampling series. The present study is not confined only to the setting of continuous (or uniformly continuous) functions, but it is also extended to the case of functions belonging to Orlicz spaces. Such spaces have been introduced in the 30s thanks to the Polish mathematician W. Orlicz, as a natural extension of L p spaces, and other useful spaces very used in Functional Analysis and its applications, such as interpolation and exponential spaces. Thus, in this paper we provide a unifying theory, not only in the sense that Durrmeyer sampling type series represent a generalization of the above sampling type operators, but also since the main convergence results will be given in the general setting of Orlicz spaces. From the literature, it is well-known that the classical Bernstein polynomials The literature about this operator and its generalizations is very wide; we quote here e.g. [27,26,29,28]. In this paper we apply the Durrmeyer method to the sampling series in a generalized form, considering operators of the following type, for w > 0, in which we replace the integral means by a general convolution integral. In fact, in the Durrmeyer sampling type series, the kernel functions ϕ and ψ satisfy certain moments conditions, together with suitable singularity assumptions, in both continuous and discrete form.
The central goal of this paper is to provide theoretical results on Durrmeyer sampling type series, starting from pointwise and uniform convergence of S ϕ,ψ w f to f on R, assuming f continuous or uniformly continuous and bounded. In this regard, we also investigate the problem of the order of the uniform convergence, in case of uniformly continuous and bounded functions, and in the latter setting, we estimate the order of approximation by means of a quantitative estimate, using the first order modulus of continuity of f . The qualitative rate of convergence is also deduced assuming f in suitable Lipschitz classes. Further, we also study the problem of the convergence of S ϕ,ψ w f to f in the general setting of Orlicz spaces. Here, we consider the most natural notion of convergence, that is the so-called "modular convergence", introduced by the modular functional defined on the space. One of the main advantage in studying approximation theorems in Orlicz spaces is the possibility to approximate not-necessarily continuous signals. This is what usually occurs in real world applications (see, e.g., [3]), in which signals are not very regular (as happens, e.g., for images). In the context of Orlicz spaces, we firstly prove a modular inequality for the operators S ϕ,ψ w , and so, we establish a modular convergence theorem. At the end of the paper, several examples of kernels ϕ and ψ have been provided together with numerical examples and graphical representations.

Preliminaries and notations
We denote by C(R) the space of all uniformly continuous and bounded functions f : R → R, by C c (R) the subspace of C(R) whose elements have compact support. Moreover by M (R) we denote the space of all (Lebesgue) measurable real functions over R.
Let us consider the functional As it is well-known (see, e.g., [35,39,8]), I ϕ is a convex modular functional on M (R) and the Orlicz space generated by ϕ is defined by The Orlicz space L ϕ (R) is a vector space and the vector subspace is called the space of all finite elements of L ϕ (R). In general E ϕ (R) is a proper subspace of L ϕ (R) and they coincide if ϕ satisfies the well-known ∆ 2 -condition, i.e., if there exists a constant M > 0 such that Examples of functions ϕ satisfying the ∆ 2 -condition are ϕ(u) = u p , 1 ≤ p < +∞, or ϕ α,β (u) = u α log β (e + u), for α ≥ 1 and β > 0, which generate respectively, the L p -spaces and the Zygmund spaces L α log β L. On the other hand, the ϕ-function ϕ α (t) = e t α − 1, α > 0, generates the so-called exponential spaces, which are examples of Orlicz spaces for which E ϕα (R) ⊂ L ϕα (R).
In L ϕ (R) we work with a notion of convergence called modular convergence: we will say that a net of functions (f w ) w>0 ⊂ L ϕ (R) is modularly convergent to a function for some λ > 0. This notion induces a topology in L ϕ (R), called modular topology.
In the space L ϕ (R) we can also introduce a norm (the Luxemburg norm), defined by Thus, we also have a stronger notion of convergence in L ϕ (R), namely the norm convergence. It is well known that f w − f ϕ → 0, as w → +∞, if and only if I ϕ [λ(f w − f )] → 0, as w → +∞, for every λ > 0. The two notions of convergence are equivalent if and only if the function ϕ satisfies the ∆ 2 -condition. For further details in the matter, see, e.g., [36,33,38,39,8].

The generalized Durrmeyer sampling series
Here we recall the definition of the family of the generalized Durrmeyer sampling operators. Such operators have been firstly introduced in [7] in order to study asymptotic expansion and Voronovskaja-type theorems in case of sufficiently regular functions. Let us consider two functions ϕ, ψ ∈ L 1 (R), such that ϕ is bounded in a neighborhood of the origin, and satisfying k∈Z ϕ(u − k) = 1, for every u ∈ R, and R ψ(u)du = 1.
Note that, ψ defines an approximate identity (see, e.g., [13,6,41,34]) by the formula ψ w (u) := wψ(wu), u ∈ R and w > 0. For any ν ∈ N 0 , let us define the discrete and continuous algebraic moments of ϕ and ψ respectively, as follows respectively. Note that, for a function f : R → R, the definition of the moments M ν (f ),M ν (f ) can also be given for any ν ≥ 0. Now, we will call kernels a pair of functions ϕ and ψ belonging to L 1 (R), satisfying (1), and such that, there exists r > 0 for which M r (ϕ) < +∞. For w > 0 and for kernels ϕ and ψ, we define a family of operators S ϕ,ψ for any given function f such that the above series is convergent, for every x ∈ R. S ϕ,ψ w are called the Durrmeyer sampling operators based on ϕ and ψ.
In order to study convergence results for the Durrmeyer sampling operators, we first show the following lemma.
Lemma 3.1. Under the above assumptions on the kernel ϕ, we have For a proof of Lemma 3.1, see, e.g., [4].

Convergence theorems
From now on, in the whole paper we will always consider kernels ϕ and ψ satisfying the assumptions introduced in Section 3. Note that, with the name kernels, we refer to both the functions ϕ and ψ, even if they satisfy different assumptions. Now, we prove the following pointwise and uniform convergence theorem.
at any point x of continuity of f . Moreover, if f ∈ C(R), then Proof. We only prove the second part of the theorem, since the first part can be obtained by similar methods. Let ε > 0 be fixed. Then there exists δ > 0 such that |f (x) − f (y)| < ε when |x − y| < δ. Let x ∈ R be fixed. Using (1), we have The first term can be further divided into Thus, Now, by the change of variable wu − k = y, and recalling that ψ ∈ L 1 (R), we have for every w ≥ 0. Thus, Moreover, Then there existsw 1 ∈ R such that By similar reasoning, we obtain the following inequality From property (ii) of Lemma 3.1, there existsw 2 ∈ R such that Finally, observing that the above estimate does not depend on x ∈ R, we easily obtain and thus the proof follows by the arbitrariness of ε > 0.
In order to obtain a modular convergence theorem in L η (R), we will study a modular continuity property for the family of Durrmeyer operators (S ϕ,ψ w ) w>0 . From now on, we denote by η a convex ϕ-function. Now, we can prove the following.
Theorem 4.2. Let ψ be a kernel such that M 0 (ψ) < +∞, and f ∈ L η (R) be fixed. Then there exists λ > 0 such that In particular, S ϕ,ψ w f is well-defined and belongs to L η (R), for every w > 0.
Applying Jensen inequality twice, the change of variable wu − k = t and Fubini-Tonelli theorem, we obtain with the change of variable wx − k = y.
As a consequence of previous theorem, it turns out that the operators S ϕ,ψ w are welldefined in L η (R) and map L η (R) into itself. Moreover, we also have that S ϕ,ψ w is modularly continuous, i.e., for any modularly convergent sequence Indeed, it is wellknown that there exists λ > 0 such that I η [λ(f − f k )] → 0, as k → +∞, and so, choosing λ > 0 such that λM 0 (ϕ)M 0 (ψ) ≤ λ, we have: as k → +∞. Now, we are able to prove the main theorem of this section.
Theorem 4.3. Let ψ be a kernel such that M 0 (ψ) < +∞, and let f ∈ L η (R) be fixed. Then there exists λ > 0 such that Proof. First of all, since f ∈ L η (R), we have that there exist λ 1 , λ 2 > 0 such that i.e., for every fixed ε > 0 there exists δ > 0 such that for every h ∈ R such that |h| ≤ δ (see, e.g., [8]). Now, we fix λ > 0 such that Thus, by the properties of the convex modular functional I η , we can write what follows: First, we estimate J 1 . Applying Jensen inequality twice similarly to the proof of Theorem 4.2, the change of variable wx − k = t and Fubini-Tonelli theorem, we obtain Now, using δ given in (4), we can split the above integral as follows Now, using the inequality in (4) with h = t w , we have For what concerns J 1,2 , by the convexity of η, we have Now we can observe that, since ϕ ∈ L 1 (R), there exists w 1 > 0 such that |t|>δw |ϕ(t)| dt < ε, for every w ≥ w 1 . Moreover, noting that for every t ∈ R and w > 0, we have for every w ≥ w 1 . Now, we estimate J 2 . By the change of variable t = u − k w , applying Jensen inequality twice and Fubini-Tonelli theorem, we have where we have used the change of variable y = wt. Then, using again δ given in (4), we can rewrite the above integral as follows: Thus, similarly to before, using the inequality in (4) with h = y w , we obtain for every w > 0. Now, for what concerns the last term J 2,2 , by the convexity of η, we have Now, we can observe that, since ψ ∈ L 1 (R), there exists w 2 > 0 such that |y|>δw |ψ(y)| dy < ε, for every w ≥ w 2 and, similarly to before, we have for every w ≥ w 2 . Finally, setting w := max {w 1 , w 2 } and we have I η λ S ϕ,ψ w f − f ≤ Kε, for every w ≥ w. Thus, the proof follows by the arbitrariness of ε.

Quantitative estimates
Here we provide a quantitative estimate for the rate of convergence of the Durrmeyer sampling operators for f ∈ C(R), in terms of the modulus of continuity, defined by We recall the following well-known inequality ω(f, λδ) ≤ (λ + 1)ω(f, δ), for every δ, λ > 0.
We can prove what follows.

Applications to particular cases
In this section we want to show how the Durrmeyer sampling type series generalize some other well-known families of sampling type series. Moreover, we will also consider applications to some special instances of Orlicz spaces.
In order to show that the generalized sampling type series, introduced by Butzer in the 80s (see, e.g., [40,11,12,14,44,2,1]), are particular cases of the Durrmeyer sampling type series, we need to give a distributional interpretation of the above operators, choosing, e.g., as kernel ψ the Dirac delta distribution δ. Indeed, using the scaling and convolution property of the Dirac delta distribution, and recalling that δ is even, the generalized sampling operators can be obtained as follows for any f ∈ C(R). Thus, S ϕ,δ w f (x) = (G ϕ w f )(x), for every x ∈ R and w > 0.
Similarly to what has been made for the generalized sampling operators, also the sampling Kantorovich operators (see, e.g., [4]) can be viewed as Durrmeyer sampling type operators.
Finally we will apply the previous convergence results in some useful cases of Orlicz spaces. First we consider the particular case when η(u) = u p for u ≥ 0 and 1 ≤ p < +∞. Here L η (R) = E η (R) = L p (R), 1 ≤ p < +∞ and in this frame, the modular convergence and the usual Luxemburg norm-convergence are equivalent. From the theory developed in the previous sections, we have the following corollaries. Corollary 6.1. Let ψ be such that M 0 (ψ) < +∞. Then, for every f ∈ L p (R), 1 ≤ p < +∞, we have In particular, S ϕ,ψ w f is well-defined in L p (R) and S ϕ,ψ w f ∈ L p (R) whenever f ∈ L p (R).

Proof.
A direct application of Theorem 4.2 with η(u) = u p , yields from which the assertion follows.
Moreover we immediately obtain the following convergence result.
As another important case, we can consider the function η α,β (u) = u α log β (e+u), u ≥ 0 for α ≥ 1 and β > 0. The corresponding Orlicz spaces are the so-called interpolation spaces and are given by the set of functions f ∈ M(R) for which for some λ > 0, and they are denoted by L α log β L(R). Note that the function η α,β satisfies the ∆ 2 -property, which means that L α log β L(R) coincides with the space of its finite elements E η α,β (R). As a consequence of the Theorem 4.2, we can obtain the following corollary, e.g. for the case α = β = 1.
Since in the above case of Orlicz spaces the ∆ 2 -condition is fulfilled, the modular convergence and the norm convergence are equivalent and we immediately obtain the following convergence theorem. Corollary 6.4. Let ψ be such that M 0 (ψ) < +∞. For every f ∈ L log L and for every λ > 0, we have where · L log L is the Luxemburg norm associated to I η 1,1 .
As last particular case, we consider the exponential spaces generated by the ϕ-function η α (u) = e u α − 1, u ≥ 0 for some α > 0. Here the Orlicz space L ηα (R) consists of those functions f ∈ M(R) for which for some λ > 0. Since η α does not satisfy the ∆ 2 -property, the space L ηα (R) does not coincide with the space of its finite elements E ηα (R). As a consequence, modular convergence is no more equivalent to norm convergence. By Theorem 4.2, we can obtain the following. Corollary 6.5. Let ψ be such that M 0 (ψ) < +∞. For every f ∈ L ηα (R), there holds In particular, S ϕ,ψ w f is well-defined in L ηα (R) and S ϕ,ψ w f ∈ L ηα (R) whenever f ∈ L ηα (R).
Since in this case ∆ 2 -property is not fulfilled, we can only state a result on modular convergence rather than on norm convergence. The next corollary follows immediately from Theorem 4.3.

Examples with graphical representations
In this last section we want to show specific examples of kernel functions ϕ and ψ, for which the results proved in this paper hold, together with some graphical examples.
The Fourier transform of σ n is given by (see, e.g., [42,14,16]), where the sinc-function is defined by The functions σ n are bounded on R for all n ∈ N with compact support [−n/2, n/2]. This implies that σ n ∈ L 1 (R) and the moment condition M r (ϕ) < +∞ is satisfied for all r > 0. It is well-known that the singularity assumption (1) on ϕ is equivalent to prove the following condition expressed in terms of σ n : The equivalence between the two conditions is a direct consequence of the Poisson summation formula (see, e.g., [13]). Rewriting explicitly the expression in (6), we have L p -space. Hence, choosing where σ 2 is the central B-spline of order 2 (see Figure 4) and ψ(t) = χ [0,1] (t), t ∈ R, we want to apply the Durrmeyer sampling series S σ 2 ,χ [0,1] w to two different discontinuous functions (see Figure 5), namely, f 1 (x) := 1, |x| ≤ 1, 0, |x| > 1, as well f 2 , defined by −50 x 4 , x ≥ 1.  In general, we want to underline that from the properties of the kernel ϕ = σ n and since ψ = χ [0,1] (which has compact support) satisfies trivially the condition (c) of Remark 3.1, Corollary 6.2, Corollary 6.4 and Corollary 6.6 hold. The Durrmeyer type sampling series with w = 5 and w = 10 of the functions f 1 and f 2 are given in Figure 6 and Figure 7, respectively. As before, the red dotted lines represent the graphs of the operators S   In conclusion, in order to underline that the convergence results proved in this paper hold for a large class of kernels ϕ and ψ, we observe that it is possible to provide examples of Durrmeyer sampling operators based on a more general kernel ψ, also with unbounded support. For example, we can choose as ψ the Fejér kernel (see Figure 8), defined by F (t) := 1 2 sinc 2 t 2 , t ∈ R. Obviously, F is bounded and non-negative on R, belongs to L 1 (R) and satisfies R F (t)dt = 1. Moreover, the moment condition M 0 (ψ) < +∞ is trivially fulfilled in view of Remark 3.1 (c) with 0 < ν < 1. Finally, it is interesting to observe that the Fejér kernel can be chosen also as the kernel ϕ. Indeed, since its Fourier transform is given by (see, e.g., [13]), it follows, by the equivalent condition (7) (applied to F in place of σ n ), that F satisfies the discrete singularity assumption (1) on ϕ.