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17 pages, 303 KiB

Open AccessArticle

Korovkin-Type Theorems for Positive Linear Operators Based on the Statistical Derivative of Deferred Cesàro Summability

by Hari Mohan Srivastava, Bidu Bhusan Jena, Susanta Kumar Paikray and Umakanta Misra

Algorithms 2025, 18(4), 218; https://doi.org/10.3390/a18040218 - 11 Apr 2025

Viewed by 497

In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish a novel Korovkin-type theorem for a specific set of exponential test functions, namely [...] Read more.

In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish a novel Korovkin-type theorem for a specific set of exponential test functions, namely 1,

e^{- υ}

and

e^{- 2 υ}

, which are defined on the Banach space

C [0, \infty)

. Our results significantly extend several well-known Korovkin-type theorems. Additionally, we analyze the rate of convergence associated with the statistical derivatives under deferred Cesàro summability. To support our theoretical findings, we provide compelling numerical examples, followed by graphical representations generated using MATLAB software, to visually illustrate and enhance the understanding of the convergence behavior of the operators. Full article

(This article belongs to the Section Algorithms for Multidisciplinary Applications)

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15 pages, 2229 KiB

Open AccessArticle

Convergence on Kirk Iteration of Cesàro Means for Asymptotically Nonexpansive Mappings

by Lale Cona and Deniz Şimşek

Symmetry 2025, 17(3), 393; https://doi.org/10.3390/sym17030393 - 5 Mar 2025

Viewed by 718

Abstract

This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable [...] Read more.

This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable norms. The focus is to determine the conditions under which the Kirk iteration sequence converges strongly or weakly to a fixed point. Finally, some examples are given in this article to demonstrate the advantages of the preferred iteration method and to verify the results obtained. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

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15 pages, 327 KiB

Open AccessArticle

Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition

by Alexander Zhdanok

Mathematics 2023, 11(15), 3388; https://doi.org/10.3390/math11153388 - 2 Aug 2023

Cited by 2 | Viewed by 1316

Abstract

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions [...] Read more.

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition

(D)

for the ergodicity of a Markov chain is equivalent to condition

(*)

: all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case. Full article

(This article belongs to the Section D1: Probability and Statistics)

12 pages, 283 KiB

Open AccessArticle

Modulated Lacunary Statistical and Strong-Cesàro Convergences

by María del Pilar Romero de la Rosa

Symmetry 2023, 15(7), 1351; https://doi.org/10.3390/sym15071351 - 3 Jul 2023

Cited by 2 | Viewed by 1045

Abstract

Here, we continued the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan which relate different convergence methods involving the classical statistical and the classical strong Cesàro convergences by means of lacunary sequences and measures of density in

N

modulated by a modulus [...] Read more.

Here, we continued the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan which relate different convergence methods involving the classical statistical and the classical strong Cesàro convergences by means of lacunary sequences and measures of density in

N

modulated by a modulus function f. A method for constructing non-compatible modulus functions was also included, which is related to symmetries with respect to

y = x

. Full article

(This article belongs to the Special Issue Advances in Matrix Transformations, Operators and Symmetry)

25 pages, 349 KiB

Open AccessArticle

Matrix Summability of Walsh–Fourier Series

by Ushangi Goginava and Károly Nagy

Mathematics 2022, 10(14), 2458; https://doi.org/10.3390/math10142458 - 14 Jul 2022

Cited by 14 | Viewed by 1548

Abstract

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in

L_{1}

space and in

C_{W}

space in terms of modulus of continuity and matrix transform variation. Moreover, we show the [...] Read more.

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in

L_{1}

space and in

C_{W}

space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator

t^{*} (f)

of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms

t_{n} (f)

of the Walsh–Fourier series are convergent almost everywhere to the function f. The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters. Full article

(This article belongs to the Section E6: Functional Interpolation)

23 pages, 368 KiB

Open AccessArticle

Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

by Han Feng and Yan Ge

Mathematics 2022, 10(12), 2108; https://doi.org/10.3390/math10122108 - 17 Jun 2022

Cited by 2 | Viewed by 1559

Abstract

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on

R^{d}

. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the [...] Read more.

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on

R^{d}

. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function

h_{κ} (x) : = \prod_{ν \in R +} {| ⟨x, ν⟩ |}^{κ_{ν}}, x \in R^{d}

on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro

(C, δ)

means in the weighted

L^{p}

space for

δ

above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

13 pages, 292 KiB

Open AccessArticle

The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

by Jeong-Gyoo Kim

Mathematics 2021, 9(4), 389; https://doi.org/10.3390/math9040389 - 15 Feb 2021

Cited by 5 | Viewed by 2277

Abstract

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of [...] Read more.

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences. Full article

16 pages, 326 KiB

Open AccessEditor’s ChoiceArticle

On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure

by M. Marin, S. Vlase, R. Ellahi and M.M. Bhatti

Symmetry 2019, 11(7), 863; https://doi.org/10.3390/sym11070863 - 2 Jul 2019

Cited by 124 | Viewed by 4189

Abstract

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based [...] Read more.

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem. Full article

(This article belongs to the Special Issue Symmetry in Applied Continuous Mechanics)

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