Approximation results in Orlicz spaces by modified sampling Kantorovich operators

Metin Turgay

doi:10.1515/dema-2025-0222

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Approximation results in Orlicz spaces by modified sampling Kantorovich operators

Metin Turgay

Veröffentlicht/Copyright: 21. Januar 2026

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Aus der Zeitschrift Demonstratio Mathematica Band 59 Heft 1

Abstract

This paper deals with the modified sampling Kantorovich operators. We start by presenting the primary notations of Orlicz spaces and fundamental auxiliary results. To show modular convergence of the corresponding operators in Orlicz spaces, we obtain well definiteness in Orlicz spaces and norm convergence in the space of continuous functions with compact support. In the last section, we present some examples of ρ-kernels which satisfy the corresponding assumptions and present some graphical representations.

Keywords: sampling Kantorovich operators; modified sampling operators; Orlicz spaces; modular convergence

MSC 2020: 41A25; 41A35; 94A20

1 Introduction

Bernstein polynomials play a significant role in approximation theory, primarily because they provide a constructive proof of the classical Weierstrass approximation theorem (see [1]). In order to enhance the rate of convergence to the desired function and reduce the approximation error, King [2] proposed a modification to the traditional Bernstein polynomials for functions f ∈ C[0, 1]. This approach involves a sequence of continuous functions r n defined on [0,1], where each r _n(x) satisfies 0 ≤ r _n(x) ≤ 1 for all x ∈ [0, 1] and n ∈ N . Later, in [3], the authors introduced another variation of the Bernstein operators for f ∈ C[0, 1] as follows:

B n f ◦ ρ − 1 ρ x = ∑ k = 0 n f ◦ ρ − 1 k n n k ρ x k 1 − ρ x n − k

where a particular function ρ : [ 0,1 ] → R is utilized under certain appropriate conditions.

On the other hand, to establish an approximation scheme valid over the entire real line, Butzer and his research group in Aachen introduced the concept of generalized sampling operators, defined by

G w χ f x := ∑ k ∈ Z f k w χ w x − k , x ∈ R , w > 0 ,

where χ : R → R denotes a specific function that fulfills certain conditions, and f : R → R is a bounded and continuous function on R . A broad range of works has examined these generalized sampling operators and their variants; see [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. To deal with the approximation of functions that are not necessarily continuous, Bardaro et al. [26] developed an L ¹-based version of these sampling operators by substituting the pointwise samples f k w with the average values w ∫ k / w ( k + 1 ) / w f u d u . This led to the formulation of sampling Kantorovich operators, expressed as

K w χ f x := ∑ k ∈ Z χ w x − k w ∫ k / w ( k + 1 ) / w f u d u , x ∈ R ,

where f is a locally integrable function and χ serves as a kernel function. The sampling Kantorovich operators have been extensively investigated across various functional settings; see [27], [28], [29], [30], [31], [32], [33].

In a very recent study [34], the authors proposed a modified version of the generalized sampling series, given by

G w χ , ρ f x = G w χ f ◦ ρ − 1 ρ x := ∑ k ∈ Z f ◦ ρ − 1 k w χ w ρ x − k , x ∈ R , w > 0 ,

where χ is an appropriately chosen kernel function. In their work, they explored the approximation capabilities of this operator in both the space of continuous functions and in weighted spaces of functions. Moreover, by employing a Voronovskaja-type result (see [34], Theorem 5]), they compared the approximation efficiency of the classical generalized sampling operators with that of their newly defined operators and demonstrated that the latter offers improved approximation under suitable conditions.

Inspired by the effectiveness of this new framework and the widespread use of sampling Kantorovich operators, the same group of authors extended their approach in [35] by introducing a modified version of the sampling Kantorovich operators, defined as

K w χ , ρ f x = K w χ f ◦ ρ − 1 ρ x := ∑ k ∈ Z χ w ρ x − k w ∫ k / w ( k + 1 ) / w f ◦ ρ − 1 u d u , x ∈ R , w > 0 ,

where χ : R → R is a suitable kernel and f ◦ ρ − 1 : R → R is a locally integrable function. They analyzed the approximation features of these operators in both standard continuous function spaces and weighted spaces of continuous functions.

This paper is structured as follows: Section 2 outlines the essential notations and preliminaries. Section 3 provides a summary of the operators along with some fundamental results. In Section 4, we focus on investigating the approximation behavior of the corresponding operators within Orlicz spaces. Final section is devoted to present some examples of ρ-kernels which satisfy corresponding assumptions and present some graphical representations.

2 Basic notations and preliminaries

We use the symbols N , Z , and R to denote the sets of positive integers, integers, and real numbers, respectively. The set of all non-negative real numbers is denoted by R 0 + . The space C R refers to the set of all continuous functions defined on R , which are not necessarily bounded. The subset C B R consists of all functions in C R that are bounded, equipped with the norm  ‖ f ‖ : = sup x ∈ R f x . Additionally, we denote by U C R the space of functions in C B R that are uniformly continuous, and by C c R the space of functions with compact support. The symbol M R stands for the set of all (Lebesgue) measurable real or complex functions, while L ∞ R represents the space of essentially bounded measurable functions on R .

Let ρ : R → R be a strictly increasing function that satisfies the following conditions:

(ρ ₁) ρ belongs to C R ;
(ρ ₂) ρ(0) = 0 and lim_x→±∞ ρ(x) = ±∞;
(ρ ₃) ρ′(x) > 1 holds for every x ∈ R
(ρ ₄) ρ ′ ∈ L ∞ R .

Remark 1.

It is possible to find many examples of ρ functions which satisfy the conditions (ρ ₁) − (ρ ₄). For example, we state ρ x = α x for all α > 1 and ρ x = 2 x + tanh ⁡ x .

2.1 Context of Orlicz spaces

Let η : R 0 + → R 0 + be a convex η-function, that is, a function satisfying the following conditions:

i. η(0) = 0, η(u) > 0 for all u > 0, and η is non-decreasing on R 0 + ;
ii. η is convex on R 0 + .

We now define the modular functional

I η f := ∫ R η f ( x ) d x f ∈ M ( R ) .

It is well known that I ^η is a convex modular on the space M ( R ) (see [36], [37], [38]), and it generates the Orlicz space defined by

L η ( R ) := f ∈ M ( R ) : I η [ λ f ] < + ∞ for some λ > 0 .

The Orlicz space L η ( R ) is a vector space, and its subspace

E η ( R ) := f ∈ M ( R ) : I η [ λ f ] < + ∞ for every λ > 0

is called the space of all finite elements of L η ( R ) . In general, E η ( R ) is a proper subspace of L η ( R ) ; however, the two spaces coincide if and only if the so-called Δ₂-condition is satisfied, that is,

η ( 2 u ) ≤ M η ( u ) , u ∈ R 0 +

for some constant M > 0. Examples of η-functions satisfying the Δ₂-condition can be found in [26],36],37].

In the context of the L η ( R ) space, we can consider two distinct notions of convergence for net of functions. The first, named “modular convergence” describes the asymptotic behavior of a sequence ( f w ) w > 0 within L η ( R ) approaching a limit function f through the modular functional I ^η. Namely, ( f w ) w > 0 is said to modularly convergent to f if for some λ > 0,

lim w → + ∞ I η [ λ ( f w − f ) ] = 0

holds. This notion induces a topology in L η ( R ) , known as the “modular topology”.

Furthermore, another concept of convergence, called “norm convergence” can be established by introducing a norm on L η ( R ) , referred as the Luxemburg norm. This norm, denoted by ‖ ⋅‖_η, is defined as

f η := inf { λ > 0 : I η [ f / λ ] ≤ 1 } .

As a result, norm convergence represents a stronger notion of convergence than modular convergence in L η R (convergence in the sense of the norm  ⋅ η generated by η of a sequence (f _n) to f implies its modular convergence to f). Easily, for a net of functions ( f w ) w > 0 in L η ( R ) , the convergence lim_w→+∞‖f _w − f‖_η = 0 is equivalent to the condition lim_w→+∞ I ^η[λ(f _w − f)] = 0 for all λ > 0. However, these two notions of convergence become coincide only if the function η satisfies the Δ₂ condition. For further details concerning these spaces, see, e.g., [36], [37], [38], [39], [40].

3 Construction of the operators and some basic results

For any β ≥ 0, absolute moment of order β associated with ρ (or simply ρ-absolute moment) is given by

M β ρ χ := sup u ∈ R ∑ k ∈ Z χ ρ u − k k − ρ u β , u ∈ R .

Definition 1.

Throughout the paper, a function χ : R → R will be called a kernel associated with ρ (or simply ρ-kernel) if it satisfies the following assumptions:

(χ1) χ ∈ C R ;
(χ2) for every u ∈ R
m 0 ρ χ , u = ∑ k ∈ Z χ ρ u − k = 1 ;
(χ3) for β > 0, ρ-absolute moment of χ is finite, that is
M β ρ χ = sup u ∈ R ∑ k ∈ Z χ ρ u − k k − ρ u β < + ∞ .

By ψ, we will denote the class of all functions satisfying the assumptions (χ1), (χ2) and (χ3). Now, we recall a family of Kantorovich type sampling operators so called modified sampling Kantorovich series given by

(3.1) K w χ , ρ f x = K w χ f ◦ ρ − 1 ρ x := ∑ k ∈ Z χ w ρ x − k w ∫ k / w ( k + 1 ) / w f ◦ ρ − 1 u d u , x ∈ R , w > 0

for χ ∈ ψ and locally integrable functions f◦ρ ⁻¹.

Now, we state a remark which was proved in [26] for classical moments of kernel functions but it is trivial to adapt to ρ-moments of kernel functions:

Remark 2.

For all functions χ belongs to ψ, we have

i. M 0 ρ χ < + ∞ ;
ii. for every δ > 0
lim w → + ∞ ∑ k − w ρ x ≥ w δ χ w ρ x − k = 0

uniformly with respect to x ∈ R ;
iii. For every δ > 0 and ɛ > 0, there exists a constant M > 0 such that
∫ ρ x > M w χ w ρ x − k d x < ε

for every sufficiently large w > 0 and k / w ∈ − δ , δ .

We also note that, for ν, γ > 0 with ν < γ , M γ ρ ( χ ) < + ∞ implies M ν ρ ( χ ) < + ∞ . When χ has compact support, we immediately have that M γ ρ ( χ ) < + ∞ for every γ > 0, see [41].

Remark 3

([35]).

i. The operator (3.1) is well-defined if, for example, f ∈ L ∞ R .
ii. Let χ ∈ ψ be a ρ-kernel. If f : R → R is a bounded function, then
lim w → ∞ K w χ , ρ f t = f t

holds at each continuity point t ∈ R of f. Moreover, if f ◦ ρ − 1 ∈ U C R , then we have
lim w → + ∞ ‖ K w χ , ρ f − f ‖ ∞ = 0 .

The proof of above items could be found in the corresponding reference.

Lemma 1.

Let f ∈ L η R . Then, we have f ◦ ρ − 1 ∈ L η R .

Proof.

By the assumption f ∈ L η R , we know that for some λ > 0, I ^η[λf] < + ∞. We aim to show that

I η λ f ◦ ρ − 1 = ∫ R η λ f ( ρ − 1 ( x ) ) d x < + ∞ .

Using change of variables x = ρ(t), we can write

I η λ f ◦ ρ − 1 = ∫ R η | λ f ( t ) | ρ ′ ( t ) d t .

Since ρ ′ ∈ L ∞ ( R ) , let C ̃ := ‖ ρ ′ ‖ ∞ < + ∞ . Then, by assumption

I η λ f ◦ ρ − 1 ≤ C ̃ ∫ R η ( | λ f ( t ) | ) d t = C ̃ I η [ λ f ] < + ∞ .

Thus, we have f ◦ ρ − 1 ∈ L η ( R ) .□

Lemma 2.

Let χ ∈ ψ be a ρ-kernel. Then, χ ∈ L 1 R .

Proof.

As in [6], (2.3)], we have with a: = ρ ⁻¹(0) and b: = ρ ⁻¹(1)

□

4 Convergence of K w χ , ρ in L η R

In this section, we present convergence of K w χ , ρ in L η R . Firstly, we start with the well-definiteness of the operators.

Theorem 1.

Let f ∈ L η R and χ ∈ ψ be a ρ-kernel. Then the operator K w χ , ρ maps L η R in L η R . Indeed,

I η λ K w χ , ρ f ≤ 1 M 0 ρ χ χ 1 I η λ M 0 ρ χ f ◦ ρ − 1

for some λ > 0.

Proof.

By Lemma 1, we have f ◦ ρ − 1 ∈ L η R . This means that, there exist λ ̃ > 0 such that I η λ ̃ f ◦ ρ − 1 < + ∞ . Choosing λ > 0 such that λ M 0 ρ χ ≤ λ ̃ , then applying Jensen’s inequality twice, we get

I η λ K w χ , ρ f = ∫ R η λ K w χ , ρ f x d x ≤ ∫ R η λ ∑ k ∈ Z χ w ρ x − k w ∫ k / w ( k + 1 ) / w f ◦ ρ − 1 u d u d x ≤ 1 M 0 ρ χ ∑ k ∈ Z ∫ k / w ( k + 1 ) / w η λ M 0 ρ χ f ◦ ρ − 1 u d u ∫ R w χ w ρ x − k d x .

Now, using change of variable w ρ x − k = v in the last integral and (ρ ₃), we get

I η λ K w χ , ρ f ≤ 1 M 0 ρ χ ∑ k ∈ Z ∫ k / w ( k + 1 ) / w η λ M 0 ρ χ f ◦ ρ − 1 u d u χ 1 ≤ 1 M 0 ρ χ χ 1 I η λ M 0 ρ χ f ◦ ρ − 1 < 1 M 0 ρ χ χ 1 I η λ ̃ f ◦ ρ − 1 < + ∞

which is desired.□

Theorem 2.

Let f ◦ ρ − 1 ∈ C c R , χ ∈ ψ be a ρ-kernel. Then,

(4.1) lim w → + ∞ K w χ , ρ f − f η = 0 .

Proof.

In order to establish the limit in (4.1), it suffices to show that for each fixed λ > 0

lim w → + ∞ ∫ R η λ K w χ , ρ f ( x ) − f ( x ) d x = 0 ,

or, equivalently, that the family of functions

α w ( x ) = η λ | K w χ , ρ f ( x ) − f ( x ) |

converges to zero in L 1 ( R ) . To prove this, we invoke the Vitali convergence theorem by verifying the following three properties:

(V ₁) α w w > 0 converges in measure to zero;
(V ₂) For every ɛ > 0 there exists a measurable set E ε ⊂ R with |E _ɛ| < ∞ such that, for any measurable F disjoint from E _ɛ,
∫ F α w ( x ) d x < ε for all sufficiently large w .
(V ₃) For each ɛ > 0 there is a δ > 0 so that whenever E ⊂ R is measurable with |E| < δ,
∫ E α w ( x ) d x < ε for all sufficiently large w .

Having established (V ₁)–(V ₃), Vitali’s theorem guarantees that α _w → 0 in L 1 ( R ) , which completes the proof of (4.1). (V ₁) follows directly from Remark 3 (ii) and continuity of η. To verify property (V ₂), fix ɛ > 0 and choose γ ̃ > 0 and γ > γ ̃ + 1 / w so that

supp f ⊂ B ( 0 , γ ̃ ) .

Observing that k w , k + 1 w ∩ B 0 , γ ̃ = ∅ whenever k∉B(0, wγ), we can write

∫ k / w k + 1 / w | f ( u ) | d u = 0 .

Next, by Remark 2 (iii) there exists M > 0 (which we may take, w.l.o.g, larger than γ ̃ ) such that for every sufficiently large w and for all k ∈ B(0, wγ) one has

∫ | ρ x | > M w χ w ρ ( x ) − k d x < ε .

By Jensen’s inequality and Fubini-Tonelli theorem, we can write what follows:

∫ ρ x > M η λ K w χ , ρ x d x ≤ ∑ k ≤ w γ η λ M 0 ρ χ f ∞ M 0 ρ χ w ∫ ρ x > M w χ w ρ x − k d x < ε ⋅ η λ M 0 ρ χ f ∞ M 0 ρ χ w ⋅ L ,

where L is the number of integers k such that k ≤ w γ . For every w ≥ 1, we have

L ≤ 2 γ w + 1 =: L ′ ,

where ⋅ denotes the integer part. Thus,

∫ ρ x > M η λ K w χ , ρ x d x < ε ⋅ η λ M 0 ρ χ f ∞ M 0 ρ χ w ⋅ L ′ =: ε ⋅ C

for every w ≥ 1. Therefore, for ɛ > 0 we set E ̃ ε = B 0 , M . Then for every measurable set F ̃ , with F ̃ ∩ E ̃ ε = ∅ , we have

∫ F ̃ η λ K w χ , ρ x − f x d x = ∫ F ̃ η λ K w χ , ρ x d x ≤ ∫ ρ x > M η λ K w χ , ρ x d x < ε ⋅ C .

Finally, concerning (V ₃), for any fixed ɛ > 0, there is a measurable set B _ɛ with

B ε < ε M 0 ρ χ η λ f ∞

for f ∞ > 0 . Then, using Remark 3 we have for every w > 0

∫ B ε η λ K w χ , ρ f x d x ≤ ∫ B ε η λ M 0 ρ χ f ∞ < ε .

This means that, the integrals

∫ ⋅ η λ K w χ , ρ f x − f x d x

are equi-absolutely continuous. Since λ > 0 is arbitrary, we have desired.□

As a final and the main result of this section, we mention the modular convergence of the operators K w χ , ρ .

Theorem 3.

f ∈ L η R and χ ∈ ψ be a ρ-kernel. Then, there exist λ > 0 such that

lim w → + ∞ I η λ K w χ , ρ f − f = 0 .

Proof.

Using Lemma 1 we know that both f and f◦ρ ⁻¹ belong to L η R . Let ɛ > 0 be fixed. By density of C c R in L η R , there exist λ′ and function g ∈ C c R such that

(4.2) I η λ ′ f − g < ε ρ ′ ∞

Let g ̃ := g ◦ ρ − 1 . It is easy to see that g ̃ is continuous and has compact support because of properties of ρ. So we can write f ◦ ρ − 1 − g ̃ = f − g ◦ ρ − 1 . Using change of variable and ρ 4 , we have

I η λ ′ f ◦ ρ − 1 − g ̃ = I η λ ′ f − g ◦ ρ − 1 = ∫ R η λ ′ f − g t ρ ′ t d t ≤ ρ ′ ∞ I η λ ′ f − g < ε .

Now, let us set λ > 0 such that 3 λ M 0 ρ χ ≤ λ ′ . Considering properties of η, (4.2) and Theorem 1, we get

I η λ K w χ , ρ f − f ≤ I η 3 λ K w χ , ρ f − K w χ , ρ g + I η 3 λ K w χ , ρ g − g + I η 3 λ f − g ≤ 1 M 0 ρ χ χ 1 I η λ ′ f − g ◦ ρ − 1 + I η 3 λ K w χ , ρ g − g + I η 3 λ f − g < 1 M 0 ρ χ χ 1 + 1 ρ ′ ∞ ε + I η 3 λ K w χ , ρ g − g .

The assertion follows from Theorem 2 for enough large w.□

5 Examples of ρ-kernels and graphical representations

In this section, we provide concrete examples of ρ-kernels that satisfy the assumptions outlined in Definition 1. We then present graphical comparisons to illustrate the approximation behavior of the modified sampling Kantorovich operators K w χ , ρ versus the classical operators K w χ .

5.1 Examples of ρ-kernels

A crucial step in applying the theory is finding suitable kernel functions. The condition χ 2 , also known as the partition of unity condition, can be conveniently verified using the Fourier transform and the Poisson summation formula (see [4], p. 123 and Section 5.1.5], [42]). Using Poisson summation formula χ 2 is equivalent to

χ ̃ 2 π k = 1 , k = 0 0 , k ∈ Z \ { 0 }

where χ ̂ v = ∫ R χ u e − i v u d u , v ∈ R .

5.1.1 The central B-spline kernel

A highly effective and widely used family of kernels are the central B-splines of order n ∈ N , defined by:

σ n ( t ) := 1 ( n − 1 ) ! ∑ j = 0 n ( − 1 ) j n j n 2 + t − j + n − 1 , t ∈ R

where (t)₊: = max{t, 0}.

These functions are ideal because they have compact support [−n/2, n/2], which guarantees that the moment condition (χ3) holds for any β > 0. The partition of unity condition χ 2 is also satisfied.

Modified sampling Kantorovich operators of f ∈ L η R takes on the form

K w σ n , ρ f x = ∑ k ∈ Z σ n w ρ x − k w ∫ k / w ( k + 1 ) / w f ◦ ρ − 1 u d u , x ∈ R , w > 0

using B-Spline kernel.

5.1.2 The Fejér kernel

Another important example is the Fejér kernel, which is not compactly supported but decays quickly. It is defined as:

F ( t ) := 1 2 sinc 2 t 2 , t ∈ R

This kernel also satisfies all the necessary assumptions to be a ρ-kernel, see [26].

Modified sampling Kantorovich operators of f ∈ L η R takes on the form

K w F , ρ f x = ∑ k ∈ Z F w ρ x − k w ∫ k / w ( k + 1 ) / w f ◦ ρ − 1 u d u , x ∈ R , w > 0

using Fejér’s kernel.

5.2 Graphical comparisons

This subsection provides a comparative analysis of the modified and classical sampling Kantorovich operators. The comparison is illustrated through graphical representations, which are generated using the central B-spline kernel of order 5. Throughout the examples, we consider the function ρ ̄ : R → R defined as ρ ̄ ( t ) = 2 t + tanh t . It is easy to see that function ρ ̄ is continuous on R , ρ ̄ 0 = 0 and lim t → ± ∞ ρ ̄ t = ± ∞ . The derivative of ρ ̄ , ρ ̄ ′ t = 2 + sech 2 t and its range is (2,3]. So, ρ ̄ ′ t > 1 for all t ∈ R and ρ ̄ ′ ∈ L ∞ R . Hence, the conditions ρ 1 − ρ 4 are all satisfied. We also consider functions f : R → R as

f ( x ) = − 1 , if x < − 2 x + 2.5 , if − 2 ≤ x < − 1 0 , if − 1 ≤ x < 0 cos π x , if 0 ≤ x ≤ 1 1 , otherwise

and g : R → R as

g ( t ) = − 1 , if t < 1 5 t 2 , otherwise

In Figure 1, the graph of function f is presented. In Figure 2, we compare classical sampling Kantorovich operators and modified sampling Kantorovich operators using w = 7 and 5th B-Spline kernel.

Figure 1:

Graph of function f.

$Figure 2: Graph of function f and operators K 7 σ 5 f , K 7 σ 5 , ρ ̄ f $\left({K}_{7}^{{\sigma }_{5}}f\right),\left({K}_{7}^{{\sigma }_{5},\bar{\rho }}f\right)$ .$

Figure 2:

Graph of function f and operators K 7 σ 5 f , K 7 σ 5 , ρ ̄ f .

In Figure 3, the graph of function g is presented. In Figure 4, we compare classical sampling Kantorovich operators and modified sampling Kantorovich operators using w = 4 and Fejer’s kernel.

Figure 3:

Graph of function g.

$Figure 4: Graph of function f and operators K 4 F g , K 4 F , ρ ̄ g $\left({K}_{4}^{F}g\right),\left({K}_{4}^{F,\bar{\rho }}g\right)$ .$

Figure 4:

Graph of function f and operators K 4 F g , K 4 F , ρ ̄ g .

Corresponding author: Metin Turgay, Department of Mathematics, Faculty of Science, Selcuk University, Selcuklu, 42003, Konya, Türkiye; and Constructive Mathematical Analysis Research Laboratory (CMARL), Selcuk University, Selcuklu, 42003, Konya, Türkiye, E-mail: metinturgay@yahoo.com

Acknowledgements

The author have been supported within TUBITAK (The Scientific and Technological Research Council of Turkey) 1001-Project 123F123. This study was also partially supported by Selcuk University Coordinator Ship of Scientific Research Project under the grant number 25401195. The author is thankful to the Reviewers for the critical remarks, which improve the exposition. This paper is dedicated to my wife Zehra Turgay and our newborn son Tuna Turgay.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Research funding: The APC of this article was supported by Selcuk University Scientific Research Coordinatorship with project number 25601096.
Data availability: All data generated or analyzed during this study are included in this published article.

References

[1] S. N. Bernstein, Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilités, Math. Charkow 13 (1912), no. 1–2.Suche in Google Scholar

[2] J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar. 99 (2003), 203–208. https://doi.org/10.1023/a:1024571126455.10.1023/A:1024571126455Suche in Google Scholar

[3] D. Cárdenas-Morales, P. Garrancho, and I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (2011), 158–163.10.1016/j.camwa.2011.04.063Suche in Google Scholar

[4] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation I, Academic Press, New York-London, 1971.10.1007/978-3-0348-7448-9Suche in Google Scholar

[5] P. L. Butzer, W. Engels, S. Ries, and R. L. Stens, The shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM J. Appl. Math. 46 (1986), no. 2, 299–323. https://doi.org/10.1137/0146020.Suche in Google Scholar

[6] S. Ries and R. L. Stens, Approximation by generalized sampling series, Proceedings of the International Conference on Constructive Theory of Functions (Varna, 1984), 1984, Sofia: Bulgarian Academy of Science, pp. 746–756.Suche in Google Scholar

[7] L. Angeloni, D. Costarelli, and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Fenn. Math. 43 (2018), no. 2, 755–767. https://doi.org/10.5186/aasfm.2018.4343.Suche in Google Scholar

[8] T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, and G. Vinti, Convergence of generalized sampling series in weighted spaces, Demonstr. Math. 55 (2022), 153–162. https://doi.org/10.1515/dema-2022-0014.Suche in Google Scholar

[9] B. R. Draganov, A fast converging sampling operator, Constr. Math. Anal. 5 (2022), no. 4, 190–201. https://doi.org/10.33205/cma.1172005.Suche in Google Scholar

[10] T. Acar and B. R. Draganov, A strong converse inequality for generalized sampling operators, Ann. Funct. Anal. 13 (2022), 36. https://doi.org/10.1007/s43034-022-00185-6.Suche in Google Scholar

[11] M. Turgay and T. Acar, Approximation by bivariate generalized sampling series in weighted spaces of functions, Dolomites Res. Notes Approx. 16 (2023), 11–22.Suche in Google Scholar

[12] D. Özer, M. Turgay, and T. Acar, Approximation properties of bivariate sampling Durrmeyer series in weighted spaces of functions, Adv. Stud. Euro-Tbil. Math. J. 16 (2023), no. Supp. 3, 89–107.10.32513/asetmj/1932200823307Suche in Google Scholar

[13] S. Kursun and T. Acar, Approximation of discontinuous signals by exponential-type generalized sampling Kantorovich series, Math. Methods Appl. Sci. 48 (2024), 1–16. https://doi.org/10.1002/mma.10330.Suche in Google Scholar

[14] S. Kursun, A. Aral, and T. Acar, Approximation results for Hadamard-type exponential sampling Kantorovich series, Mediterr. J. Math. 20 (2023), 263. https://doi.org/10.1007/s00009-023-02459-2.Suche in Google Scholar

[15] S. Kursun, M. Turgay, O. Alagoz, and T. Acar, Approximation properties of multivariate exponential sampling series, Carpathian Math. Publ. 13 (2021), no. 3, 666–675.10.15330/cmp.13.3.666-675Suche in Google Scholar

[16] A. Aral, T. Acar, and S. Kursun, Generalized Kantorovich forms of exponential sampling series, Anal. Math. Phys. 12 (2022), 50. https://doi.org/10.1007/s13324-022-00667-9.Suche in Google Scholar

[17] O. Alagoz, M. Turgay, T. Acar, and M. Parlak, Approximation by sampling durrmeyer operators in weighted space of functions, Numer. Funct. Anal. Optim. 43 (2022), no. 10, 1223–1239. https://doi.org/10.1080/01630563.2022.2096630.Suche in Google Scholar

[18] T. Acar, S. Kursun, and O. Acar, Approximation properties of exponential sampling series in logarithmic weighted spaces, Bull. Iranian Math. Soc. 50 (2024), 36. https://doi.org/10.1007/s41980-024-00868-x.Suche in Google Scholar

[19] S. Kursun, A. Aral, and T. Acar, Riemann–Liouville fractional integral type exponential sampling Kantorovich series, Expert Systems with Applications 238 (2024), no. F, 122350. https://doi.org/10.1016/j.eswa.2023.122350.Suche in Google Scholar

[20] T. Acar, A. Eke, and S. Kursun, Bivariate generalized Kantorovich-type exponential sampling series, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118 (2024), 35. https://doi.org/10.1007/s13398-023-01535-2.Suche in Google Scholar

[21] T. Acar, S. Kursun, and M. Turgay, Multidimensional Kantorovich modifications of exponential sampling series, Quaest. Math. 46 (2023), no. 1, 57–72. https://doi.org/10.2989/16073606.2021.1992033.Suche in Google Scholar

[22] O. Alagoz, Weighted approximations by sampling type operators: recent and new results, Constr. Math. Anal. 7 (2024), no. 3, 114–125. https://doi.org/10.33205/cma.1528004.Suche in Google Scholar

[23] D. Costarelli and A. R. Sambucını, A comparison among a fuzzy algorithm for image rescaling with other methods of digital image processing, Constr. Math. Anal. 7 (2024), no. 2, 45–68.10.33205/cma.1467369Suche in Google Scholar

[24] L. Boccali, D. Costarelli, and G. Vinti, A Jackson-type estimate in terms of the τ-modulus for neural network operators in Lp-spaces, Modern Math. Methods 2 (2024), no. 2, 90–102. https://doi.org/10.64700/mmm.42.Suche in Google Scholar

[25] L. Angeloni, D. D. Bloisi, P Burghignoli, D Comite, D Costarelli, M Piconi, et al.., Microwave remote sensing of soil moisture, above ground biomass and freeze-thaw dynamic: modeling and empirical approaches, Modern Math. Methods 3 (2025), no. 2, 57–71. https://doi.org/10.64700/mmm.54.Suche in Google Scholar

[26] C. Bardaro, P. L. Butzer, R. L. Stens, and G. Vinti, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (2007), no. 1, 29–52. https://doi.org/10.1007/bf03549462.Suche in Google Scholar

[27] C. Bardaro and I. Mantellini, Asymptotic formulae for multivariate Kantorovich type generalized sampling series, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 7, 1247–1258. https://doi.org/10.1007/s10114-011-0227-0.Suche in Google Scholar

[28] C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim. 33 (2012), no. 4, 374–396. https://doi.org/10.1080/01630563.2011.652270.Suche in Google Scholar

[29] D. Costarelli, A. M. Minotti, and G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl. 450 (2017), no. 2, 1083–1103. https://doi.org/10.1016/j.jmaa.2017.01.066.Suche in Google Scholar

[30] T. Acar, D. Costarelli, and G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series, Banach J. Math. Anal. 14 (2020), no. 4, 1481–1508. https://doi.org/10.1007/s43037-020-00071-0.Suche in Google Scholar

[31] M. Cantarini, D. Costarelli, and G. Vinti, Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators, J. Math. Anal. Appl. 509 (2022), 125913. https://doi.org/10.1016/j.jmaa.2021.125913.Suche in Google Scholar

[32] T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, and G. Vinti, Approximation by sampling Kantorovich series in weighted spaces of functions. Turkish J. Math. 46 (2022), no. 7, 2663–2676. https://doi.org/10.55730/1300-0098.3293.Suche in Google Scholar

[33] T. Acar and B. R. Draganov, A characterization of the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification, J. Math. Anal. Appl. 530 (2024), no. 2, 127740. https://doi.org/10.1016/j.jmaa.2023.127740.Suche in Google Scholar

[34] M. Turgay and T. Acar, Approximation by modified generalized sampling series, Mediterr. J. Math. 21 (2024), 107. https://doi.org/10.1007/s00009-024-02653-w.Suche in Google Scholar

[35] M. Turgay and T. Acar, Modified sampling kantorovich operators in weighted spaces of functions, FILOMAT 39 (2025), no. 24, 8477–8491.Suche in Google Scholar

[36] C. Bardaro, J. Musielak, and G. Vinti, Nonlinear Integral Operators and Applications, de Gruyter Series in Nonlinear Analysis and Applications, vol. 9, Walter de Gruyter & Co., Berlin, 2003.10.1515/9783110199277Suche in Google Scholar

[37] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.10.1007/BFb0072210Suche in Google Scholar

[38] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991.Suche in Google Scholar

[39] J. Musielak and W. Orlicz, On modular spaces, Studia Math. 28 (1959), 49–65. https://doi.org/10.4064/sm-18-1-49-65.Suche in Google Scholar

[40] M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002.Suche in Google Scholar

[41] D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integral Equations Appl. 26 (2014), no. 4, 455–481. https://doi.org/10.1216/jie-2014-26-4-455.Suche in Google Scholar

[42] P. L. Butzer, W. Splettstosser, and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Dtsch. Math.-Ver. 90 (1988), 1–70.Suche in Google Scholar

Received: 2025-08-29

Accepted: 2025-12-12

Published Online: 2026-01-21

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https://doi.org/10.1515/dema-2025-0222

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sampling Kantorovich operators; modified sampling operators; Orlicz spaces; modular convergence

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