The analogs of the Korovkin theorems in banach function spaces

Zeren, Yusuf; Исмайлов, М. И.; Karaçam, Cemil

doi:10.1007/s11117-022-00897-y

“…Other results related to the rate of convergence of Kantorovich operators in Lebesgue spaces can be found in [1,3,7]. we give an example for Theorem 3.5 where the exact rate of convergence of Kantorovich operators in Lebesgue spaces is attained.…”

Section: Resultsmentioning

confidence: 77%

Revisiting Kantorovich Operators in Lebesgue Spaces

Obie,

Taebenu,

Gunadi

et al. 2023

JIMS

0

View full text Add to dashboard Cite

According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorovich operators as certain modifications for Bernstein polynomials. In this paper, we investigate the behaviour of Kantorovich operators in Lebesgue spaces. We first give an alternative proof of the uniform boundedness of Kantorovich operators in Lebesgue spaces by using the Riesz-Thorin Interpolation Theorem. In addition, we examine the convergence of Kantorovich operators in the space of essentially bounded functions. We also give an example related to the rate of convergence of Kantorovich operators in a subspace of Lebesgue spaces.

show abstract

“…Observe that Theorem 1.2 can be viewed as a complement to [2, Theorem 1.4] where the assumptions in [2, Theorem 1.4] are q > 1 and f ′ belongs to M p q ([0, 1]). We also refer the reader to [9] for the extension of the case q > 1 of Theorem 1.2 to more general Banach function spaces, where the authors used a different proof to that of Theorem 1.2. Although the result in [9] covers many function spaces, it does not cover the case q = 1 of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…We also refer the reader to [9] for the extension of the case q > 1 of Theorem 1.2 to more general Banach function spaces, where the authors used a different proof to that of Theorem 1.2. Although the result in [9] covers many function spaces, it does not cover the case q = 1 of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

On the convergence of Kantorovich operators in Morrey spaces

Gunadi

¹

,

Hakim

²

,

Sawano

³

2023

Positivity

0

View full text Add to dashboard Cite

In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy-Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam.

show abstract

On the convergence of Kantorovich operators in Morrey spaces

Gunadi

¹

,

Hakim

²

,

Sawano

³

2022

Preprint

View full text Add to dashboard Cite

In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy-Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam. MSC (2020): 41A10, 41A25, 42B35, 47B38

show abstract

The analogs of the Korovkin theorems in banach function spaces

Cited by 3 publications

References 24 publications

Revisiting Kantorovich Operators in Lebesgue Spaces

Revisiting Kantorovich Operators in Lebesgue Spaces

On the convergence of Kantorovich operators in Morrey spaces

On the convergence of Kantorovich operators in Morrey spaces

Contact Info

Product

Resources

About