Quantitative estimates for nonlinear sampling Kantorovich operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of continuity in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $L^{p}$-spaces, $1\leq p<\infty $, and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming $f$ in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $L^p$-case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

The linear version of the sampling Kantorovich type operators has been first introduced in [7] in one-dimensional setting and there, some approximation results in the setting of Orlicz spaces L ϕ (R) have been achieved.
It is well-known that, Orlicz spaces are very general spaces including, among its various special cases, the L p -spaces (see e.g. [35,36,8]). Subsequently, these operators were extended in [39,40] to the nonlinear case. The order of approximation for nonlinear sampling Kantorovich operators have been studied in [22] considering functions in suitable Lipschitz classes both in the space of uniformly continuous and bounded functions and in Orlicz spaces. Results concerning the multidimensional version of these operators have been obtained in [24].
In the last forty years, the study of approximation results by sampling-type operators (in linear and nonlinear cases) has been a wide research area both red from a theoretical and an application point of view, such as signal and image processing. In particular, sampling type operators (in their multivariate version) can be used in order to reconstruct and approximate images, see e.g. [37,29,18,19].
Concerning the problem of the order of approximation for the (linear) sampling Kantorovich operators, a quantitative estimate in the setting of Orlicz spaces in terms of modulus of continuity has been very recently established in [26]. On the other hand, quantitative estimates with respect to the Jordan variation for sampling-type operators have been obtained in [2] exploiting a suitable modulus of smoothness for the space of absolutely continuous functions AC (R). However, a quantitative approach for nonlinear sampling Kantorovich operators has not been addressed as yet.
In the present paper, we prove some quantitative estimates for the nonlinear sampling Kantorovich operators using the modulus of continuity of L ϕ (R). Further, the qualitative order of approximation is established for functions belonging to suitable Lipschitz classes. In the particular case of L papproximation, we directly established a quantitative estimate for the order of approximation, with the main purpose to obtain a sharper estimate than that one achieved in the general case. If the latter estimated is applied for the linear version of the sampling Kantorovich operators, we become able to improve the result that could be derived from Theorem 3.1 of [26]. Finally, we give some concrete examples of nonlinear sampling Kantorovich operators constructed by using Fejér and Bspline kernels, establishing some particular results in these instances.

Preliminaries
In this section, we recall the necessary background material related to Orlicz spaces used throughout the paper.
We denote by C (R) the space of all uniformly continuous and bounded functions f : R → R endowed with the norm · ∞ . Also, C c (R) is the subspace of C (R) consisting of functions with compact support and M (R) is the linear space of Lebesgue measurable functions f : R → R (or C).
Let us introduce the functional I ϕ associated to the ϕ−function ϕ and defined by for every f ∈ M (R) . It is well-known that I ϕ is a modular functional (see, e.g., [36,8]) and the Orlicz space generated by ϕ is defined by In general E ϕ (R) is a proper subspace of L ϕ (R) and they coincide if and only if the so-called ∆ 2 −condition on ϕ is satisfied, i.e. there exists a constant M > 0 such that Examples of ϕ−functions satisfying ∆ 2 −condition are ϕ (u) = u p for 1 ≤ p < ∞ (in this case L ϕ (R) = L p (R)) or ϕ α,β (u) = u α ln β (e + u) for α ≥ 1, β > 0 (in this case, the Orlicz spaces L α,β (R) is the interpolation space L α log β L (R)). A concept of convergence in Orlicz spaces, called modular convergence, was introduced in [35]. We say that a net of functions (f w ) w>0 ⊂ L ϕ (R) is modularly convergent to f ∈ L ϕ (R) , if there exists λ > 0 such that Now, we can recall the definition of the modulus of continuity in Orlicz spaces L ϕ (R) , with respect to the modular I ϕ .
For any fixed f ∈ L ϕ (R) and for a suitable λ > 0, we denote with δ > 0.
In order to recall the class of operators we work with, we need some additional concepts.
Let Π := (t k ) k∈Z be a sequence of real numbers such that −∞ < t k < t k+1 < +∞ for every k ∈ Z, lim k→±∞ t k = ±∞ and there are two positive constants ∆, δ such that δ ≤ ∆ k := t k+1 − t k ≤ ∆, for In what follows, a function χ : R × R → R will be called a kernel if it satisfies the following conditions: (χ1) k → χ (wx − t k , u) ∈ ℓ 1 (Z), for every (x, u) ∈ R 2 and w > 0; (χ2) χ (x, 0) = 0, for every x ∈ R; (χ3) χ is a (L, ψ)-Lipschitz kernel, i.e., there exist a measurable function L : R → R + 0 and a ϕ-function ψ, such that for every x, u, v ∈ R; (χ4) there exists θ 0 > 0 such that as w → +∞, uniformly with respect to x ∈ R. Moreover, we assume that the function L of condition (χ3) satisfies the following properties: (L1) L ∈ L 1 (R) and is bounded in a neighborhood of the origin; (L2) there exists β 0 > 0 such that i.e., the absolute moment of order β 0 is finite. Then, nonlinear sampling Kantorovich operators for a given kernel χ are defined by where f : R → R is a locally integrable function such that the above series is convergent for every x ∈ R.
The following lemma will be needed in the proof of our main theorems.

Main Results
In this section, firstly we give the following quantitative estimate for nonlinear sampling Kantorovich operators by using the modulus of continuity in Orlicz spaces. For this aim, we need a growth condition on the composition of the function ϕ, which generates the Orlicz space and the function ψ of condition (χ3).
In correspondence to λ, by condition ( uniformly with respect to x ∈ R, for sufficiently large w > 0. Let us choose µ > 0 such that Taking into account that ϕ is convex and non-decreasing, for µ > 0, we can write Now, we estimate I 1 . Applying condition (χ3), we have Applying Jensen inequality twice (see e.g., [20]), the change of variable y = x − t k /w, condition (H) and Fubini-Tonelli theorem, we obtain Now, let 0 < α < 1 be fixed. We now split the above integral J as For J 1 , one has On the other hand, taking into account that I η is convex (since η is convex), for J 2 we can write Moreover, it can be easily seen that for every y. Therefore, by assumption (3.1), we have Now we estimate I 2 .
Using condition (χ3), the change of variable y = u − t k /w, Jensen inequality and condition (H), we have Using Jensen inequality and Fubini-Tonelli theorem, we get For I 3 , denoted by A 0 ⊆ R the set of all points of R for which f = 0 almost everywhere, we obtain By the convexity of ϕ and condition (χ4), we have for positive constants M 2 , θ 0 . This completes the proof.
Note that, condition (3.1) is satisfied when, for instance, the kernel χ satisfies condition (χ3) with L having compact support, e.g.
We now recall the definition of Lipschitz classes in Orlicz spaces. We define by Lip ϕ (ν), 0 < ν ≤ 1, the set of all functions f ∈ M (R) such that, there exists λ > 0 with: as t → 0. In this context, from Theorem 3.1 we immediately get the following corollary.
From the theory developed in [39], we know that if the function ψ of condition (χ3) is ψ(u) = u, u ∈ R, and ϕ(u) = u p , 1 ≤ p < ∞, the operators S w maps the whole space L p (R) into itself and therefore, we can obtain, as particular case, a quantitative estimate in L p (R). But since the ϕ-modulus of continuity does not satisfy the well-known property ω(f, λδ) ≤ (1 + λ)ω(f, δ), satisfied e.g., by the ω p -modulus of smoothness below defined, we can proceed using a direct approach and estimating the term S w f − f with respect to the p-norm. For the above purpose, we need to recall the definition of the L p −modulus of smoothness of order one given as We can prove the following estimate.
Proof. Recalling that I ϕ [f ] = f p p , when ϕ(u) = u p , proceeding as in the first part of the proof of Theorem 3.1, using the Minkowsky inequality, and that the function | · | 1/p is concave and hence sub-additive, we immediately obtain: We estimate I 1 . Applying Jensen inequality twice, Fubini-Tonelli theorem, and by the change of variable y = x − t k /w, we get for every w > 0, where L 1 and M p (L) are both finite. Note that, in the above estimates we used the following well-known property of the modulus of smoothness Now we estimate I 2 . Using Jensen inequality twice, the change of variable y = u − t k /w and Fubini-Tonelli theorem, we have For I 3 , denoted by A 0 ⊆ R the set of all points of R for which f = 0 almost everywhere, we obtain From condition (χ4), we have for positive constants M 2 , θ 0 . This proves the theorem. Now, denoting by Lip (α, p), 0 < α ≤ 1, p ≥ 1, the corresponding Lipschitz classes in L p (R), we can immediately state the following.
for every sufficiently large w > 0, where m 0,Π (L) is finite in view of Lemma 2.1 and, C 1 > 0, M 2 , θ 0 > 0 are the constants arising from the fact that f ∈ Lip(α, p) and from condition (χ4), respectively.
Remark 3.1. Note that if the kernel χ is of the form χ (x, u) = L (x) u, with L satisfying conditions (L1) and (L2) , the above operators reduces to the linear case considered in [7]. In this situation, condition (χ4) becomes uniformly with respect to x ∈ R, for some θ 0 > 0. Sometimes, a stronger condition (instead of (3.3)) is required, i.e., that k∈Z L (u − t k ) = 1, (3.4) for every u ∈ R. In this case, condition (χ4) holds for every θ 0 > 0. When t k = k (uniform sampling scheme) and L is continuous, it is well known that (3.4) is equivalent to where L (υ) := R L (u) e −iυu du, υ ∈ R, denotes the Fourier transform of L (see [7,11,23]).
The rate of approximation for (linear) sampling Kantorovich operators in various settings was studied in [23]. Also, a quantitative estimate for these operators was obtained in [26] by using the modulus of continuity in Orlicz spaces.
The general setting of Orlicz spaces allows us to directly deduce the results concerning some quantitative estimates of approximation in L p −spaces (as in Corollary 3.3), together with some other useful spaces, as for examples Zygmund spaces and the exponential spaces, defined in Section 2.
In the case of approximation by linear sampling Kantorovich operators, considered in Remark 3.1, and below denoted by S * w , we can immediately deduce, as a particular case, the following results. Corollary 3.4. Suppose that M p (L) < +∞, for 1 ≤ p < ∞. Then, for every f ∈ L p (R) , there holds for sufficiently large w > 0, where m 0,Π (L) < +∞. Moreover, if f ∈ Lip (α, p) , with 0 < α ≤ 1, we have for sufficiently large w > 0, where C 1 > 0 is the constant arising from the fact that f belongs to Lip(α, p).
Note that, the estimates established in Corollary 3.4 are sharper than those achieved in the general case of Theorem 3.2.

Examples of kernels
In this section, we give some concrete examples of the above nonlinear sampling Kantorovich operators describing a natural procedure to construct kernels. We will consider kernel functions of the form where (g w ) w>0 is a family of functions g w : R → R satisfying g w (u) → u uniformly on R as w → +∞ and such that there exists a ϕ−function ψ with for every u, υ ∈ R and w > 0. Hence, assumptions (χi) , i = 1, .., 4 and (Lj) , j = 1, 2 can be summarized as follows.
for every x ∈ R and w > 0, L is locally bounded in a neighborhood of the origin and there exists β 0 > 0 such that (4.2) (L2) g w (0) = 0, for every w > 0; (L3) there exists θ 0 > 0 such that as w → +∞, uniformly with respect to x ∈ R. Firstly, we show an example of sequence (g w ) w>0 satisfying all the assumptions of the above theory.
Example 4.1. Let us define g w (u) = u 1−1/w for every u ∈ (a, 1) , with 0 < a < 1/e and g w (u) = u otherwise, for w > 0. It is easily seen that g w (u) → u uniformly on R, as w → +∞. Note that if the function L satisfies condition (3.4), assumption (L3) holds for θ 0 = 1. In fact, the function g w (u)−u on (a, 1) achieves the maximum at u 0 := w−1 w w for sufficiently large w > 0 (g w (u)−u = 0 otherwise), then we have for every u ∈ R for sufficiently large w > 0 and for a suitable positive constant C. Then as w → +∞. Moreover, g w (u) satisfies (4.1) for sufficiently large w > 0 and ψ concave.
In addition, if we consider the particular case g w (u) = u, u ∈ R, w > 0, the function ψ corresponding to χ (x, u) = L (x) u is ψ (u) = u, u ∈ R + 0 . In this case, our operators reduce to linear ones studied in details in [7,23,1] and as stated in Remark 3.1, condition (L3) becomes uniformly with respect to x ∈ R, which is fulfilled for every θ 0 > 0, if k∈Z L(u − k) = 1, for every u ∈ R.