Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces

1 Introduction

Bardaro et al. [1] first introduced the sampling Kantorovich (SK) operators as a development of the generalized sampling operators (e.g., [2,3]), and later, a wide research has been conducted by some authors with the aim to create some generalizations of these operators, making them more suitable to different fields of applications (e.g., [4–8]). It is important to note that one of the main advantages in considering SK-type operators resides in the possibility to approximate/reconstruct signals which are not necessarily continuous. It is worth noting that all these sampling operators are different generalizations of the celebrated Wittaker-Kotelnikov-Shannon sampling theorem (e.g., [9]).

Currently, the study of these operators and their variants (e.g., [10–15]) continues to be an active area of research in the field of signal processing, since these operators are largely useful for the applications in image reconstruction and enhancement, as for instance, in medicine (e.g., [16,17]).

Over the past few years, SK operators have been studied from many theoretical points of view: the convergence has been examined in [18] in the general context of modular spaces, while other approximation results have been investigated in [19,20].

The order of approximation for these operators has been widely examined in terms of the modulus of smoothness. Several works can be recalled in this context, as for instance, [5,21–24] for SK operators, [25,26] for Durrmeyer-type operators and [27] for Steklov sampling operators.

In real-world scenarios, signals are often affected by various types of noise. To adapt these operators to such situations, a version of the SK operators perturbed by multiplicative noise has been introduced in [28]. Convergence results for these perturbed operators have been established in Orlicz spaces in [28] and in modular spaces in [29].

However, the rate of convergence for these operators in Orlicz spaces remains an open problem and this is the main focus of the present work.

In this study, we establish quantitative estimates for the order of approximation of the above perturbed operators; to do this, we exploit the definition of the modulus of smoothness, defined by means of the modular which generates the space. This allows us to apply the above results in certain Orlicz spaces of particular interest, such as the interpolation spaces, the exponential spaces and the L p -spaces, 1 ≤ p < + ∞ . In particular, in the latter case, we also provide an estimate established using a direct proof based on certain properties of the L p -modulus of smoothness, which are not valid in the general case of Orlicz spaces. Finally, we briefly mention some examples of kernels for which the theory established in this work holds.

2 Notations

From now on, we denote with M ( R ) the space of all measurable functions and with C ( R ) the space of all uniformly continuous and bounded functions, endowed with ‖ ⋅ ‖ C ( R ) , defined as ‖ f ‖ C ( R ) ≔ sup x ∈ R ∣ f ( x ) ∣ , f ∈ C ( R ) .

We recall some basic useful notions as follows.

Now, we consider the functional I φ associated with a given φ -function φ and defined as follows:

for every f ∈ M ( R ) . The functional I φ is modular and it can be used in order to define the Orlicz space. Indeed, the Orlicz space generated by φ is

Now, we recall the notion of the modular convergence in Orlicz spaces [30–32].

Further, in order to obtain some estimates, we introduce the following definition in Orlicz spaces [30].

It is well known that, since f ∈ L φ ( R ) , there exists λ > 0 such that ω ( λ f , δ ) φ → 0 , as δ → 0 + . For further details regarding Orlicz spaces refer [30,33–35].

In order to define the operators considered in this work for any function χ : R → R , we can define the following useful tools. The discrete and the continuous absolute moments of order r ( r ≥ 0 ) of χ are defined in this way, respectively.

and

It is well known that if χ is bounded on R and χ is O ( ∣ u ∣ − r − 1 − ε ) , ∣ u ∣ → + ∞ , ε > 0 , we obtain that m β ( χ ) < + ∞ and m ˜ β ( χ ) < + ∞ , for every 0 ≤ β ≤ r .

From now on, the function χ will be called a kernel if it satisfies the following assumptions:

1. χ ∈ L 1 ( R ) is bounded in a neighbourhood of the origin;

2. for some μ > 0

   ∑ k ∈ Z χ ( w x − k ) − 1 = O ( w − μ ) , as w → + ∞

   uniformly with respect to x ∈ R ;

3. there exists β > 0 such that m β ( χ ) < + ∞ .

From the above properties, for any given kernel χ , it is possible to prove the following condition [1]:

Now, we recall the definition of the family of generalized SK operators perturbed by multiplicative noise, ( K w χ , G ) w > 0 [28], which will be considered in the following, as:

where G ≔ ( G w ) w > 0 is a family of noise sequences, with G w = ( g k , w ) k ∈ Z , g k , w : R → R + are locally integrable noise functions with ∫ k ⁄ w ( k + 1 ) ⁄ w g k , w ( u ) d u ≠ 0 , for every k ∈ Z and w > 0 , and f : R → R such that g k , w f are locally integrable and the above series is convergent for every x ∈ R .

3 Quantitative estimate in Orlicz spaces

In this section, we prove a quantitative estimate for the perturbed SK operators, by using the modulus of smoothness of L φ ( R ) .

From now on, we require that the φ -function is convex and that the following hypothesis on the boundedness of the noise functions g k , w is true:

There exist positive constants δ , σ such that

In order to ensure that the operators are well-defined in L φ ( R ) , we recall the following result.

Now, we can establish the following estimate in Orlicz spaces.

In case the kernel χ satisfies the strong condition (2.2) instead of ( χ 2 ) , we can deduce the following corollary.

We note that assumption (3.1) is satisfied, for instance, by kernels having a sufficiently rapid decay, as kernels χ with compact support. Indeed, if we suppose that supp χ ⊂ [ − B , B ] , B > 0 , we can write

for sufficiently large w > B 1 ⁄ ( 1 − α ) and 0 < α < 1 . So, Theorem 3.2 can be reformulated in this way.

If the kernel χ does not have compact support, we may require this assumption on the continuous absolute moment

Under this assumption, we obtain that condition (3.1) is true with γ = ( 1 − α ) q and K = m ˜ q ( χ ) and so the thesis of Theorem 3.2 holds. For more details and for examples of kernels χ with unbounded support satisfying (3.1), refer, e.g., [24,36].

In order to deduce also the quantitative order of approximation of the function f through the operators ( K w χ , G f ) w > 0 , we recall the definition of the Lipschitz class in Orlicz spaces, denoted by Lip φ ( ν ) , 0 < ν ≤ 1 and defined as follows:

From the previous result, we can immediately deduce the following:

In the literature, there are many examples of remarkable Orlicz spaces. Indeed, the φ -function defined as φ ( u ) ≔ u p , 1 ≤ p < + ∞ generates the well-known L p -spaces, or with a φ -function in the form φ ( u ) ≔ u α log β ( u + e ) , for α ≥ 1 , β > 0 , we obtain the L α log β L -spaces or Zygmund spaces largely used, for instance, in the theory of partial differential equations. If instead we consider the φ -function as follows: φ ( u ) ≔ e u γ − 1 , γ > 0 , u ≥ 0 , we have the exponential spaces used in embedding theorems between Sobolev spaces. In all the above cases, the results developed here are valid.

4 Quantitative estimate in L p -spaces

We have just noted that the theory established in Section 3 continues to be satisfied in the specific case of L p -spaces. In this setting, if we consider the problem of the order of approximation by a direct approach, we can obtain a sharper estimate than the one deduced by what was achieved in Section 3. In order to do this, we will use the following well-known inequality:

that is true in L p -spaces (but not in Orlicz spaces). In order to reach this goal, we consider the definition of the modulus of smoothness in L p ( R ) :