Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces

Esma Kangal; Ülkü Dinlemez Kantar

doi:10.1515/dema-2025-0155

Article Open Access

Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces

Esma Kangal and Ülkü Dinlemez Kantar

Published/Copyright: September 20, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This study examines Durrmeyer-type exponential sampling series to obtain a quantitative estimate by using the concept of the logarithmic modulus of smoothness defined with the help of a suitable modular functional on Mellin-Orlicz spaces. Also, we want to underline that the spaces on which we study are reduced versions of Mellin-Orlicz spaces. Additionally, we obtain a further estimate for a particular case of Mellin-Lebesgue space, i.e., X 0 p , by utilizing an appropriate logarithmic modulus of smoothness that is different from the previous one. Also, we obtain direct estimates on the order of approximation for two different types of logarithmic Lipschitz classes by using previously mentioned results in involved sections.

Keywords: Mellin-Lebesgue spaces; Mellin-Orlicz spaces; logarithmic modulus of smoothness; order of approximation

MSC 2010: 41A25; 41A35; 46E30; 47A58; 47B38; 94A20

1 Introduction

Approximation theory is concerned with the existence of a more practical function that converges to a function g . The Weierstrass approximation theorem, which is the cornerstone of approximation theory, states that for each ε > 0 , there exists P n ( x ) , which is a polynomial of degree at most n such that

max x ∈ [ a , b ] ∣ P n ( x ) − g ( x ) ∣ < ε ,

for ∀ g ∈ C [ a , b ] . In [1], in order to prove it more effortlessly, Bernstein established the following sum:

( B n g ) ( x ) = ∑ k = 0 n n k x k ( 1 − x ) ( n − k ) g k n ( x ∈ [ 0 , 1 ] , n ∈ N ) ,

for a continuous function g defined on [0, 1]. Academic research has been and continues to be inspired by Bernstein polynomials (for studies in recent years see e.g., [2–7]). As can be seen, the approximation obtained by using Bernstein polynomials is provided for continuous functions. However, when it comes to discontinuous functions, the series named after Kantorovich gives approximation results for a countable discontinuous function g on [0, 1]. In [8], using the Steklov mean values ( n + 1 ) ∫ k n + 1 k + 1 n + 1 g ( u ) d u instead of the values g k n ,

Kantorovich constructed the sum as follows:

( K n g ) ( x ) = ( n + 1 ) ∑ k = 0 n n k x k ( 1 − x ) ( n − k ) ∫ k n + 1 k + 1 n + 1 g ( u ) d u , ( x ∈ [ 0 , 1 ] ) .

In [9], Durrmeyer developed the following Bernstein-Durrmeyer sum:

( D n g ) ( x ) = ( n + 1 ) ∑ k = 0 n n k x k ( 1 − x ) ( n − k ) ∫ 0 1 n k t k ( 1 − t ) ( n − k ) g ( t ) d t , ( x ∈ [ 0 , 1 ] ) ,

to obtain approximations for Lebesgue integrable functions on [0, 1].

When we leave the function spaces defined on compact intervals and turn our direction to the spaces of functions defined on the entire real axis, the studies in the field of sampling theory guide us. Whittaker, Kotelnikov [10,11], and Shannon [12] give the sampling formula for Fourier band limited functions without being aware of one another. Motivated by these studies, Butzer [13] introduced a generalization of the sampling series to obtain approximation results for the functions without requiring the functions to be Fourier band-limited. This generalization is defined as follows:

( S w ϕ g ) ( x ) = ∑ k ∈ Z g ( k w ) ϕ ( w x − k ) , w > 0 ,

where there are some specific assumptions for ϕ . Inspired by this series and the previously mentioned series, researchers have defined several operators such as the Kantorovich-type and the Durrmeyer-type sampling series and have studied their convergence properties. For examples from the last few years, see [14–25]. When we move from Fourier analysis to Mellin analysis, we encounter the exponential sampling formula, which is the analog of the sampling formula in Mellin analysis. It is introduced in [26–28]. Butzer and Jansche [29] prove that Mellin band-limited functions are reconstructed via the exponential sampling formula. We should clearly state that the set of Mellin band-limited functions and the set of Fourier band-limited functions are disjoint except for the trivial functions [30]. It is a rather restrictive condition that a function must be Mellin band-limited; to overcome this difficulty, Bardaro and his colleagues defined a generalization of exponential sampling series in [31]. For any function g making the series E w ϕ g convergent for all x ∈ R + , this generalization is defined as follows:

(1) ( E w ϕ g ) ( x ) = ∑ k ∈ Z g ( e k w ) ϕ ( e − k x w ) , ( w > 0 , x ∈ R + ) ,

where the kernel function ϕ has some certain properties. It is not always able to obtain an exact value for g ( e k w ) , in which case, in a similar way to what is done in the sampling series, by calculating the Steklov value in the interval [ k w , k + 1 w ] , Angamuthu and Bajpeyi constructed the Kantorovich-type exponential sampling series in [32] as follows:

(2) ( K w ϕ g ) ( x ) = ∑ k ∈ Z ϕ ( e − k x w ) w ∫ k w k + 1 w g ( e u ) d u , ( w > 0 , x ∈ R + ) ,

for any local integrable function g that makes the aforementioned series convergent for ∀ x ∈ R + , where the kernel function ϕ has some suitable assumptions. Bardaro and Mantellini [33] introduced the Durrmeyer-type exponential sampling series in the following form:

(3) ( D w ϕ , ψ g ) ( x ) ≔ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u ) d u u , ( w > 0 , x ∈ R + ) ,

under some certain assumptions on the kernels ϕ and ψ . Durrmeyer-type exponential sampling series in (3) has importance considering that it includes the generalized exponential sampling series in (1) and the Kantorovich-type exponential sampling series in (2). Several modifications of the series given in (1), including the Kantorovich-type and Durrmeyer-type exponential sampling series, have also been studied by researchers. For instances from the past few years, see [34–40].

Considering functions of Orlicz spaces, Costarelli and his colleagues developed a quantitative estimate for Durrmeyer-type sampling series by benefiting from an appropriate modulus of smoothness in [41]. Motivated by this article, we obtain an estimate for Durrmeyer-type exponential sampling series by taking into account the functions in Mellin-Orlicz spaces.

In Section 2, we remind the reader of some basic definitions and concepts related to the topic we study. We also announce the abbreviations used. In Section 3, we present the Durrmeyer-type exponential sampling series and the properties of its kernels. Also, we give the definitions of discrete and continuous absolute moments for a function on R + . In Section 4, we deduce an estimate for Durrmeyer-type exponential sampling series by considering Mellin-Orlicz spaces. We customize it for a logarithmic Lipschitz class. Finally, in Section 5, using a different logarithmic modulus of smoothness than before, we obtain a result for X 0 p , which is a reduced version of Mellin-Lebesgue spaces X c p by choosing c = 0 . Then, we obtain a direct result for a logarithmic Lipschitz class established by involved logarithmic modulus of smoothness.

2 Preliminaries and notations

We denote the sets of positive integers, integer numbers, positive real numbers, non-negative real numbers, and real numbers by N , Z , R + , R 0 + , and R , respectively.

Let L ∞ ( R + ) be the space of all essentially bounded functions endowed with the usual norm ‖ ⋅ ‖ ∞ .

Let us give the definition of convex ϕ -function. Let η : R 0 + → R 0 + . If the function η possesses the following properties:

η is convex in R 0 + ,
η ( 0 ) = 0 and η ( x ) > 0 for x > 0 , then the function η is considered as a convex ϕ -function.

For any measurable set S ⊂ R + , the measure defined as μ ( S ) = ∫ S d t t is called by Haar measure. Let M ( R + , μ ) denote the space of all measurable functions over R + with respect to the Haar measure μ .

For g ∈ M ( R + , μ ) , now we present the following convex modular functional:

I η [ g ] = ∫ 0 ∞ η ( ∣ g ( x ) ∣ ) d x x .

For more specific details, see [42]. The functional I η is extremely essential because it is utilized in the construction of Mellin-Orlicz space. With the help of the functional I η , the special version of Mellin-Orlicz space generated by η is defined as follows:

L η ( R + , μ ) = { g ∈ M ( R + , μ ) : I η [ λ g ] < + ∞ , for ∃ λ > 0 } .

Moreover, Mellin-Orlicz space is defined by

X c η = { g : R + → R : g ( ⋅ ) ( ⋅ ) c ∈ L η ( R + , μ ) } .

When η ( u ) = u p ( 1 ≤ p < ∞ ), the Mellin-Orlicz space X c η , indeed, is the Mellin-Lebesgue space given by X c p = { g : R + → R : g ( ⋅ ) ( ⋅ ) c ∈ L p ( R + , μ ) } , where L p ( R + , μ ) is the Lebesgue space according to the Haar measure μ . Here, it ought to emphasize that L p ( R + , μ ) is a space of functions equipped with the norm

‖ g ‖ p , μ = ∫ R + ∣ g ( u ) ∣ p d u u 1 p .

In this article, we study on the Mellin-Orlicz space X c η by choosing c = 0 . Actually, it is trivial that X 0 η = L η ( R + , μ ) . From this moment on, we no longer use X 0 η ; instead, we use just L η ( R + , μ ) .

Let us explain the notion of modular convergence. Let ( g w ) w > 0 be a net of functions that are the elements of the space L η ( R + , μ ) . For g ∈ L η ( R + , μ ) , if there exists λ > 0 such that

lim w → ∞ I η [ λ ( g w − g ) ] = 0 ,

we say that the net of functions ( g w ) is modularly convergent to g .

Here, we want to recall the definition I η -modulus of continuity in Mellin-Orlicz space L η ( R + , μ ) . It is defined by

ω η , μ ( g , δ ) = sup ∣ log t ∣ ≤ δ I η [ g ( ⋅ t ) − g ( ⋅ ) ] ,

for g ∈ L η ( R + , μ ) . From now on, we will call it by the logarithmic modulus of smoothness and we use this term throughout our study. More importantly, we note that there exists λ > 0 such that

lim δ → 0 + ω η , μ ( λ g , δ ) = 0 ,

for every g ∈ L η ( R + , μ ) . We derive this inference from [43]; also, we can see it in [42].

3 Exponential sampling Durrmeyer operator

First, we list two functions used to create these series along with their necessary properties. Let us denote the first of them by ϕ : R + → R . Let us assume that ϕ is a continuous function with the following properties:

(ϕ.1) for all u ∈ R + ,
∑ k ∈ Z ϕ ( e − k u ) = 1 ,
(ϕ.2)
M 0 ( ϕ ) ≔ sup u ∈ R + ∑ k ∈ Z ∣ ϕ ( e − k u ) ∣ < + ∞ .

Let us denote the second of these functions by ψ : R + → R . Let ψ be a function with the following properties:

(ψ.1)
∫ 0 ∞ ψ ( u ) d u u = 1 ,
(ψ.2)
M ˜ 0 ( ψ ) ≔ ∫ 0 ∞ ∣ ψ ( u ) ∣ d u u < + ∞ .

Now, we mention that the discrete absolute moments of order z ≥ 0 of ϕ are defined by

M z ( ϕ , u ) ≔ ∑ k ∈ Z ∣ ϕ ( e − k u ) ∣ ∣ log z ( e k u − 1 ) ∣ = ∑ k ∈ Z ∣ ϕ ( e − k u ) ∣ ∣ ( k − log u ) ∣ z

and

M z ( ϕ ) ≔ sup u ∈ R + ∑ k ∈ Z ∣ ϕ ( e − k u ) ∣ ∣ k − log u ∣ z .

Also, the continuous absolute moment of order z ≥ 0 of ψ is in the form

M ˜ z ( ψ ) ≔ ∫ 0 ∞ ∣ ψ ( u ) ∣ ∣ log ( u ) ∣ z d u u .

Throughout this study, we presume that ϕ and ψ satisfy the aforementioned conditions. For any w > 0 , exponential sampling Durrmeyer series are defined by

( ℰ w ϕ , ψ g ) ( x ) ≔ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u ) d u u , x ∈ R + ,

for a function g : R + → R ∈ dom ( ℰ w ϕ , ψ ) . Here, dom ( ℰ w ϕ , ψ ) denotes the domain that includes all functions that make the series convergent at each point x .

In light of [33], we can obtain the following remark for w > 0 .

Remark 1

The operators ( ℰ w ϕ , ψ ) w > 0 are well defined. Actually, we have the following inequality:

∣ ( ℰ w ϕ , ψ g ) ( x ) ∣ ≤ M 0 ( ϕ ) M ˜ 0 ( ψ ) ‖ g ‖ ∞ , x ∈ R + ,

for any g ∈ L ∞ ( R + ) .

4 Main result

We utilize the logarithmic modulus of smoothness in Mellin-Orclicz spaces to obtain an estimate (a quantitative estimate) for the Durrmeyer-type exponential sampling series. To do that, we first define the following condition which we use in hypothesis of our result.

For any 0 < α < 1 , a function ζ : R + → R satisfies the integral decay condition ( D α ) on condition that we have

w ∫ ∣ log ( u ) ∣ > 1 w α ∣ ζ ( u w ) ∣ d u u ≤ C w − c , as w → ∞ ,

for some appropriate positive constants C and c , which depend on α and ζ .

Now, we present our first main theorem.

Theorem 1

In addition to previous assumptions on the kernel functions ϕ and ψ , suppose M 0 ( ψ ) and M ˜ 0 ( ϕ ) are finite. Let η be a convex ϕ -function and g ∈ L η ( R + , μ ) . Additionally, let ϕ and ψ be kernels satisfying ( D α ) for α ∈ ( 0 , 1 ) . So, there exist two positive constants c ¯ and C ¯ , which depend on α , ϕ , and ψ , such that

I η [ λ ( ℰ w ϕ , ψ g − g ) ] ≤ M 0 ( ψ ) M ˜ 0 ( ϕ ) + M 0 ( ϕ ) M ˜ 0 ( ψ ) 2 M 0 ( ϕ ) M ˜ 0 ( ψ ) ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) , w − α ) + C ¯ M 0 ( ϕ ) + M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) I η [ 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ] w − c ¯ ,

for some λ > 0 and for every sufficiently large w > 0 . Furthermore, the preceding inequality alludes to that the Durrmeyer-type exponential sampling series ℰ w ϕ , ψ ( g ) modularly converge to g on condition that λ > 0 is sufficiently small.

Proof

Suppose that λ > 0 be fixed. We can obtain the following inequality:

I η [ λ ( ℰ w ϕ , ψ g − g ) ] = ∫ 0 ∞ η ( λ ∣ ( ℰ w ϕ , ψ g ) ( x ) − g ( x ) ∣ ) d x x

= ∫ 0 ∞ η λ ( ℰ w ϕ , ψ g ) ( x ) − ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k w ) d u u + ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k w ) d u u − g ( x ) d x x ≤ 1 2 ∫ 0 ∞ η 2 λ ( ℰ w ϕ , ψ g ) ( x ) − ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k w ) d u u d x x + ∫ 0 ∞ η 2 λ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k w ) d u u − g ( x ) d x x ≕ 1 2 { J 1 + J 2 } ,

due to η being a convex ϕ -function.

First, we estimate J 1 . After applying discrete and continuous version of the Jensen inequality, we obtain

∣ J 1 ∣ ≤ ∫ 0 ∞ η 2 λ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ g ( u ) − g ( u x e − k w ) d u u d x x ≤ 1 M 0 ( ϕ ) ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ η 2 λ M 0 ( ϕ ) w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ g ( u ) − g ( u x e − k w ) d u u d x x ≤ 1 M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ η 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ( u ) − g ( u x e − k w ) d u u d x x .

Now, let α ∈ ( 0 , 1 ) be the parameter of ( D α ) . By the change of variable y = e − k ⁄ w x and then using the Fubini-Tonelli theorem, we obtain

∣ J 1 ∣ ≤ 1 M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ 0 ∞ ∣ ϕ ( y w ) ∣ w ∫ 0 ∞ ∑ k ∈ Z ∣ ψ ( e − k u w ) ∣ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( u ) − g ( u y ) ∣ ) d u u d y y ≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ ≤ w − α + ∫ ∣ log y ∣ > w − α ∣ ϕ ( y w ) ∣ w ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( u ) − g ( u y ) ∣ ) d u u d y y ≕ J 1,1 + J 1,2 .

Let us examine J 1,1 . Using the definition of the logarithmic modulus of smoothness of L η ( R + , μ ) , we have the following inequality:

∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( u ) − g ( u y ) ∣ ) d u u = I η [ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( g ( ⋅ ) − g ( ⋅ y ) ) ] ≤ sup I η [ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( g ( ⋅ ) − g ( ⋅ y ) ) ] = ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , ∣ log y ∣ ) .

Thanks to this inequality, we can write

J 1,1 ≔ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ ≤ w − α ∣ ϕ ( y w ) ∣ w ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( u ) − g ( u y ) ∣ ) d u u d y y ≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ ≤ w − α ∣ ϕ ( y w ) ∣ w ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , ∣ log y ∣ ) d y y

≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , w − α ) ∫ ∣ log y ∣ ≤ w − α ∣ ϕ ( y w ) ∣ w d y y ≤ M 0 ( ψ ) M ˜ 0 ( ϕ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , w − α ) .

Now, let us evaluate J 1,2 . Thanks to convexity of η , we obtain

J 1,2 ≔ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ > w − α ∣ ϕ ( y w ) ∣ w ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( u ) − g ( u y ) ∣ ) d u u d y y ≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ > w − α ∣ ϕ ( y w ) ∣ w ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( ∣ g ( u ) ∣ + ∣ g ( u y ) ∣ ) ) d u u d y y ≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) ∫ ∣ log y ∣ > w − α ∣ ϕ ( y w ) ∣ w 1 2 ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( ∣ g ( u ) ∣ ) ) d u u + ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( ∣ g ( u y ) ∣ ) ) d u u d y y = M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) I η [ 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ] w ∫ ∣ log y ∣ > w − α ∣ ϕ ( y w ) ∣ d y y

because of that

∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( ∣ g ( u ) ∣ ) ) d u u = ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( ∣ g ( u y ) ∣ ) ) d u u ,

for every y ∈ R + .

In our hypothesis, we assume that the kernel ϕ satisfies integral decay condition ( D α ) ; therefore, there exist positive constants C ϕ and c ϕ such that

w ∫ ∣ log ( y ) ∣ > w − α ∣ ϕ ( y w ) ∣ d y y ≤ C ϕ w − c ϕ ,

where w is sufficiently large. Eventually, we have

J 1,2 ≤ M 0 ( ψ ) M 0 ( ϕ ) M ˜ 0 ( ψ ) I η [ 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ] C ϕ w − c ϕ ,

for sufficiently large w .

Let us continue to investigate J 2 . At this stage, we use the assumptions determined earlier for the kernels ϕ and ψ . In particular, after the change of variable t = u e − k w , applying Jensen’s inequality twice and the Fubini-Tonelli theorem once, we obtain

J 2 ≔ ∫ 0 ∞ η 2 λ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k w ) − g ( x ) d u u d x x ≤ ∫ 0 ∞ η 2 λ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ g ( u x e − k w ) − g ( x ) d u u d x x = ∫ 0 ∞ η 2 λ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( t w ) ∣ ∣ g ( x t ) − g ( x ) ∣ d t t d x x

≤ 1 M 0 ( ϕ ) ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ η 2 λ M 0 ( ϕ ) w ∫ 0 ∞ ∣ ψ ( t w ) ∣ ∣ g ( x t ) − g ( x ) ∣ d t t d x x ≤ ∫ 0 ∞ η 2 λ M 0 ( ϕ ) w ∫ 0 ∞ ∣ ψ ( t w ) ∣ ∣ g ( x t ) − g ( x ) ∣ d t t d x x ≤ 1 M ˜ 0 ( ψ ) ∫ 0 ∞ w ∫ 0 ∞ ∣ ψ ( t w ) ∣ η 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) − g ( x ) ∣ d t t d x x = w M ˜ 0 ( ψ ) ∫ 0 ∞ ∣ ψ ( t w ) ∣ ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) − g ( x ) ∣ ) d x x d t t .

Because of that

∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) − g ( x ) ∣ ) d x x = I η [ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( g ( ⋅ t ) − g ( ⋅ ) ) ] ,

we have

J 2 = w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ ≤ w − α + ∫ ∣ log t ∣ > w − α ∣ ψ ( t w ) ∣ I η [ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( g ( ⋅ t ) − g ( ⋅ ) ) ] d t t ≔ J 2,1 + J 2,2 ,

where 0 < α < 1 is the parameter of decay condition ( D α ) .

First, let us evaluate J 2,1 :

J 2,1 ≔ w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ ≤ w − α ∣ ψ ( t w ) ∣ I η [ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ( g ( ⋅ t ) − g ( ⋅ ) ) ] d t t ≤ w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ ≤ w − α ∣ ψ ( t w ) ∣ ω η , μ ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , ∣ log t ∣ ) d t t ≤ ω η , μ 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g , 1 w α

due to the inequality

w ∫ ∣ log t ∣ ≤ w − α ∣ ψ ( t w ) ∣ d t t ≤ M ˜ 0 ( ψ ) .

Second, let us find a result for J 2,2 .

J 2,2 ≔ w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ > w − α ∣ ψ ( t w ) ∣ ∫ 0 ∞ η ( 2 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) − g ( x ) ∣ ) d x x d t t ≤ w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ > w − α ∣ ψ ( t w ) ∣ 1 2 ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) ∣ ) d x x + η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x ) ∣ ) d x x d t t .

By using the following equality

∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x t ) ∣ ) d x x = ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x ) ∣ ) d x x ,

for all t ∈ R + , we have

J 2,2 ≤ w M ˜ 0 ( ψ ) ∫ ∣ log t ∣ > w − α ∣ ψ ( t w ) ∣ ∫ 0 ∞ η ( 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) ∣ g ( x ) ∣ ) d x x d t t

≤ 1 M ˜ 0 ( ψ ) I η [ 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ] w ∫ ∣ log t ∣ > w − α ∣ ψ ( t w ) ∣ d t t .

Because of our assumption that the kernel ψ satisfies ( D α ) , two positive constants c ψ and C ψ exist such that

w ∫ ∣ log ( t ) ∣ > w − α ∣ ψ ( t w ) ∣ d t t ≤ C ψ w − c ψ ,

for sufficiently large w . If we use this inequality, we obtain

J 2,2 ≤ 1 M ˜ 0 ( ψ ) I η [ 4 λ M 0 ( ϕ ) M ˜ 0 ( ψ ) g ] C ψ w − c ψ ,

where w is sufficiently large.

If we choose

C ¯ = max ( C ϕ , C ψ ) and c ¯ = min ( c ϕ , c ψ ) ,

we obtain the desired result.□

As for how to find the example kernels satisfying the decay condition ( D α ) , we can give one property (condition) given in Remark 2.

Remark 2

Let ζ : R + → R and α ∈ ( 0 , 1 ) . We suppose that β > 1 and ζ ( t ) = O ( ∣ log t ∣ − β ) , as ∣ t ∣ → ∞ , then ζ satisfies the decay condition ( D α ) , where C ≔ M β − 1 and c ≔ ( 1 − α ) ( β − 1 )

w ∫ ∣ log t ∣ > w − α ∣ ζ ( t w ) ∣ d t t ≤ M β − 1 w − ( 1 − α ) ( β − 1 ) .

Now, we introduce logarithmic Lipschitz class denoted by Lip η ( v , μ ) with 0 < v ≤ 1 . For a function g ∈ L η ( R + , μ ) , if there exists λ > 0 satisfying

I η [ λ ( g ( ⋅ t ) − g ( ⋅ ) ) ] = ∫ R + η ( λ ∣ g ( x t ) − g ( x ) ∣ ) d x x = O ( ∣ log t ∣ v ) ,

as ∣ t ∣ → 1 , we say g ∈ Lip η ( v , μ ) .

By Theorem 1, we directly obtain the following result.

Corollary 1

Suppose that g ∈ Lip η ( v , μ ) with ( 0 < v ≤ 1 ) , α ∈ ( 0 , 1 ) , and the presumptions of Theorem 1 are hold. Then, there exist M > 0 and λ > 0 such that

I η [ λ ( ℰ w ϕ , ψ g − g ) ] ≤ M w − r ,

for every sufficiently large w > 0 , where r ≔ min { α v , c ¯ } and c ¯ > 0 represent the parameter derived from Theorem 1.

5 Mellin-Lebesgue spaces L p ( R + , μ )

Here, we will obtain a quantitative estimate for the functions in L p ( R + , μ ) ( 1 ≤ p < ∞ ) . To do that, we utilize the logarithmic modulus of smoothness defined as follows:

ω p , μ ( g , δ ) = sup ∣ log t ∣ ≤ δ ∫ R + ∣ g ( x t ) − g ( x ) ∣ p d x x 1 p ( δ > 0 ) ( 1 ≤ p < ∞ ) ,

for g ∈ L p ( R + , μ ) . We obtain a sharper quantitative estimate, thanks to a property of L p ( R + , μ ) -logarithmic modulus of smoothness. This mentioned property is not true in general for ω η , μ ( g , δ ) .

Theorem 2

Suppose that both discrete and continuous absolute moments of order zero are finite for each ϕ and ψ . Furthermore, for some p with 1 ≤ p < ∞ , assume that M ˜ p ( ϕ ) and M ˜ p ( ψ ) are also finite. Then, for ∀ g ∈ L p ( R + , μ ) , we obtain the following quantitative estimate:

‖ ℰ w ϕ , ψ g − g ‖ p , μ ≤ M 0 ( ϕ ) [ 2 M ˜ 0 ( ψ ) ] p − 1 p M 0 ( ψ ) M ˜ 0 ( ϕ ) M ˜ p ( ϕ ) M 0 ( ϕ ) 1 p + ( M ˜ 0 ( ψ ) + M ˜ p ( ψ ) ) 1 p ω p , μ g , 1 w ,

for sufficiently large w.

Proof

We proceed similar to the initial part of the proof of Theorem 1. After that, we apply the Minkowsky inequality. So we obtain

‖ ℰ w ϕ , ψ g − g ‖ p , μ = ∫ 0 ∞ ∣ ( ℰ w ϕ , ψ g ) ( x ) − g ( x ) ∣ p d x x 1 p = ∫ 0 ∞ ( ℰ w ϕ , ψ g ) ( x ) − ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k ⁄ w ) d u u + ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) g ( u x e − k ⁄ w ) d u u − g ( x ) p d x x 1 p ≤ ∫ 0 ∞ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) ( g ( u ) − g ( u x e − k ⁄ w ) ) d u u p d x x 1 p + ∫ 0 ∞ ∑ k ∈ Z ϕ ( e − k x w ) w ∫ 0 ∞ ψ ( e − k u w ) ( g ( u x e − k ⁄ w − g ( x ) ) ) d u u p d x x 1 p ≔ I 1 + I 2 .

Here we evaluate I 1 : first, let us apply triangle inequality; later, Jensen inequality twice in a row. So, we have

( I 1 ) p ≤ ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ ∣ g ( u x e − k ⁄ w − g ( u ) ) ∣ d u u p d x x ≤ ( M 0 ( ϕ ) ) p − 1 ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ ∣ g ( u x e − k ⁄ w ) − g ( x ) ∣ d u u p d x x ≤ ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ ∣ g ( u x e − k ⁄ w ) − g ( u ) ∣ p d u u d x x = ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 M 0 ( ψ ) ∫ 0 ∞ w ∣ ϕ ( y w ) ∣ ∫ 0 ∞ ∣ g ( u y ) − g ( u ) ∣ p d u u d y y ,

by the change of variable e − k ⁄ w x = y and the Fubini-Tonelli theorem.

It is easy to see that

(4) ω p , μ ( g , λ δ ) ≤ ( 1 + λ ) ω p , μ ( g , δ ) ( λ , δ > 0 ) .

By using this inequality, we obtain

( I 1 ) p ≤ ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 M 0 ( ψ ) ∫ 0 ∞ w ∣ ϕ ( y w ) ∣ ω p , μ ( g , ∣ log y ∣ ) p d y y

≤ ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 M 0 ( ψ ) ω p , μ g , 1 w p ∫ 0 ∞ w ∣ ϕ ( y w ) ∣ ( 1 + ∣ w log y ∣ ) p d y y ≤ ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 M 0 ( ψ ) ω p , μ g , 1 w p 2 p − 1 ∫ 0 ∞ w ∣ ϕ ( y w ) ∣ ( 1 + ∣ w log y ∣ p ) d y y ≤ ( M 0 ( ϕ ) ) p − 1 ( M ˜ 0 ( ψ ) ) p − 1 M 0 ( ψ ) ω p , μ g , 1 w p 2 p − 1 [ M ˜ 0 ( ϕ ) + M ˜ p ( ϕ ) ] < ∞ ,

for ∀ w > 0 due to our assumptions.

Now, we observe I 2 . After substituting e − k ⁄ w u = y , let us employ the Jensen inequality and the Fubini-Tonelli theorem one after another; in this way, we yield

( I 2 ) p = ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( e − k u w ) ∣ ∣ g ( u x e − k ⁄ w ) − g ( x ) ∣ d u u p d x x = ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( y w ) ∣ ∣ g ( x y ) − g ( x ) ∣ d y y p d x x ≤ ( M 0 ( ϕ ) ) p − 1 ∫ 0 ∞ ∑ k ∈ Z ∣ ϕ ( e − k x w ) ∣ w ∫ 0 ∞ ∣ ψ ( y w ) ∣ ∣ g ( x y ) − g ( x ) ∣ d y y p d x x ≤ ( M 0 ( ϕ ) ) p ( M ˜ 0 ( ϕ ) ) p − 1 ∫ 0 ∞ w ∫ 0 ∞ ∣ ψ ( y w ) ∣ ∣ g ( x y ) − g ( x ) ∣ p d y y d x x = ( M 0 ( ϕ ) ) p ( M ˜ 0 ( ϕ ) ) p − 1 ∫ 0 ∞ w ∣ ψ ( y w ) ∣ ∫ 0 ∞ ∣ g ( x y ) − g ( x ) ∣ p d x x d y y .

Then, by employing the inequality given in (4), we finally come to the following conclusion:

( I 2 ) p ≤ 2 p − 1 ( M 0 ( ϕ ) ) p ( M ˜ 0 ( ϕ ) ) p − 1 ω p , μ g , 1 w p ∫ 0 ∞ w ∣ ψ ( y w ) ∣ ( 1 + w ∣ log y ∣ p ) d y y = 2 p − 1 ( M 0 ( ϕ ) ) p ( M ˜ 0 ( ϕ ) ) p − 1 ω p , μ g , 1 w p ( M ˜ 0 ( ψ ) + M ˜ p ( ψ ) ) < ∞ ,

for every w > 0 .□

We want to highlight that, in Theorem 2, we use the more classical assumptions M ˜ p ( ϕ ) < ∞ and M ˜ p ( ψ ) < ∞ instead of the decay condition ( D α ) . These presumptions lead us to a sharper conclusion.

Now, we give a definition of logarithmic Lipschitz class in L p ( R + , μ ) with the help of the norm ∣ ⋅ ∣ p , μ as follows:

(5) Lip ( v , μ ) p ≔ { g ∈ L p ( R + , μ ) : ‖ g ( ⋅ t ) − g ( ⋅ ) ‖ p , μ = O ( ∣ log t ∣ v ) , as ∣ t ∣ → 1 } ,

with 0 < v ≤ 1 and 1 ≤ p < ∞ .

Now, we can deduce the following result from Theorem 2.

Corollary 2

Let ϕ and ψ be kernels with M ˜ p ( ϕ ) < ∞ and M ˜ p ( ψ ) < ∞ for some 1 ≤ p < ∞ . Then, for every g ∈ Lip ( v , μ ) p ( 0 < v ≤ 1 ) , we obtain

‖ ℰ w ϕ , ψ g − g ‖ p , μ ≤ M 0 ( ϕ ) [ 2 M ˜ 0 ( ψ ) ] p − 1 p M 0 ( ψ ) M ˜ 0 ( ϕ ) + M ˜ p ( ϕ ) M 0 ( ϕ ) 1 p + ( M ˜ 0 ( ψ ) + M ˜ p ( ψ ) ) 1 p C 1 w v ,

for every sufficiently large w. Here, the constant C > 0 is derived from the definition in (5).

Examples of the kernels satisfying the prerequisites for our theorems are presented in [33].

Acknowledgments

We would like to express our sincere appreciation to Prof. Dr. Birol Altın since he thoughtfully addressed the challenges we encountered during our study and offered invaluable consultation at any time. We are also grateful to the referees for their valuable and insightful comments, which greatly helped improve the quality of this manuscript.

Funding information: Authors state no funding involved.
Author contributions: All authors accepted the responsibility for the content of the manuscript and consented to its submission, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors have no conflicts of interest to declare.
Data availability statement: Data sharing is not applicable to this article as the article describes entirely theoretical research.

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Received: 2024-12-12

Revised: 2025-05-02

Accepted: 2025-05-29

Published Online: 2025-09-20

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https://doi.org/10.1515/dema-2025-0155

Keywords for this article

Mellin-Lebesgue spaces; Mellin-Orlicz spaces; logarithmic modulus of smoothness; order of approximation

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