Symmetry

Research

15 pages, 2256 KiB

Open AccessArticle

An Accelerated Fixed-Point Algorithm with an Inertial Technique for a Countable Family of G-Nonexpansive Mappings Applied to Image Recovery

by Kobkoon Janngam and Rattanakorn Wattanataweekul

Symmetry 2022, 14(4), 662; https://doi.org/10.3390/sym14040662 - 24 Mar 2022

Cited by 3 | Viewed by 1856

Abstract

Many authors have proposed fixed-point algorithms for obtaining a fixed point of G-nonexpansive mappings without using inertial techniques. To improve convergence behavior, some accelerated fixed-point methods have been introduced. The main aim of this paper is to use a coordinate affine structure [...] Read more.

Many authors have proposed fixed-point algorithms for obtaining a fixed point of G-nonexpansive mappings without using inertial techniques. To improve convergence behavior, some accelerated fixed-point methods have been introduced. The main aim of this paper is to use a coordinate affine structure to create an accelerated fixed-point algorithm with an inertial technique for a countable family of G-nonexpansive mappings in a Hilbert space with a symmetric directed graph G and prove the weak convergence theorem of the proposed algorithm. As an application, we apply our proposed algorithm to solve image restoration and convex minimization problems. The numerical experiments show that our algorithm is more efficient than FBA, FISTA, Ishikawa iteration, S-iteration, Noor iteration and SP-iteration. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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9 pages, 270 KiB

Open AccessArticle

Yet Another New Variant of Szász–Mirakyan Operator

by Ana Maria Acu and Gancho Tachev

Symmetry 2021, 13(11), 2018; https://doi.org/10.3390/sym13112018 - 25 Oct 2021

Cited by 10 | Viewed by 2024

Abstract

In this paper, we construct a new variant of the classical Szász–Mirakyan operators,

M_{n}

, which fixes the functions 1 and

e^{a x}, x \geq 0, a \in R

. For these operators, we provide a quantitative Voronovskaya-type result. [...] Read more.

In this paper, we construct a new variant of the classical Szász–Mirakyan operators,

M_{n}

, which fixes the functions 1 and

e^{a x}, x \geq 0, a \in R

. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of

M_{n}

and a direct quantitative estimate are obtained. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Our results improve and extend similar ones on this topic, established in the last decade by many authors. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

19 pages, 521 KiB

Open AccessArticle

On a New Construction of Generalized q-Bernstein Polynomials Based on Shape Parameter λ

by Qing-Bo Cai and Reşat Aslan

Symmetry 2021, 13(10), 1919; https://doi.org/10.3390/sym13101919 - 12 Oct 2021

Cited by 21 | Viewed by 2045

Abstract

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter

λ

on the symmetric interval

[- 1, 1]

. Firstly, we computed some moments and central [...] Read more.

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter

λ

on the symmetric interval

[- 1, 1]

. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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12 pages, 294 KiB

Open AccessArticle

On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators

by Ulrich Abel and Octavian Agratini

Symmetry 2021, 13(10), 1841; https://doi.org/10.3390/sym13101841 - 1 Oct 2021

Viewed by 1589

Abstract

The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of [...] Read more.

The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube

{[0, 1]}^{q}

,

q > 1

. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

12 pages, 267 KiB

Open AccessArticle

Convergence of Certain Baskakov Operators of Integral Type

by Marius Mihai Birou, Carmen Violeta Muraru and Voichiţa Adriana Radu

Symmetry 2021, 13(9), 1747; https://doi.org/10.3390/sym13091747 - 19 Sep 2021

Cited by 2 | Viewed by 1597

Abstract

In the present paper, we propose a Baskakov operator of integral type using a function

φ

on

[0, \infty)

with the properties:

φ (0) = 0,

φ^{'} > 0

on

[0, \infty)

and [...] Read more.

In the present paper, we propose a Baskakov operator of integral type using a function

φ

on

[0, \infty)

with the properties:

φ (0) = 0,

φ^{'} > 0

on

[0, \infty)

and

{lim}_{x \to \infty} φ (x) = \infty

. The proposed operators reproduce the function

φ

and constant functions. For the constructed operator, some approximation properties are studied. Voronovskaja asymptotic type formulas for the proposed operator and its derivative are also considered. In the last section, the interest is focused on weighted approximation properties, and a weighted convergence theorem of Korovkin’s type on unbounded intervals is obtained. The results can be extended on the interval

(- \infty, 0]

(the symmetric of the interval

[0, \infty)

from the origin). Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

12 pages, 664 KiB

Open AccessArticle

Enhancement of Cone-Beam Computed Tomography Dental-Maxillofacial Images by Sampling Kantorovich Algorithm

by Danilo Costarelli, Pietro Pozzilli, Marco Seracini and Gianluca Vinti

Symmetry 2021, 13(8), 1450; https://doi.org/10.3390/sym13081450 - 9 Aug 2021

Cited by 1 | Viewed by 1819

Abstract

In this paper, we establish a procedure for the enhancement of cone-beam computed tomography (CBCT) dental-maxillofacial images; this can be useful in order to face the problem of rapid prototyping, i.e., to generate a 3D printable file of a dental prosthesis. In the [...] Read more.

In this paper, we establish a procedure for the enhancement of cone-beam computed tomography (CBCT) dental-maxillofacial images; this can be useful in order to face the problem of rapid prototyping, i.e., to generate a 3D printable file of a dental prosthesis. In the proposed procedure, a crucial role is played by the so-called sampling Kantorovich (SK) algorithm for the reconstruction and image noise reduction. For the latter algorithm, it has already been shown to be effective in the reconstruction and enhancement of real-world images affected by noise in connection to engineering and biomedical problems. The SK algorithm is given by an optimized implementation of the well-known sampling Kantorovich operators and their approximation properties. A comparison between CBTC images processed by the SK algorithm and other well-known methods of digital image processing known in the literature is also given. We finally remark that the above-treated topic has a strong multidisciplinary nature and involves concrete biomedical applications of mathematics. In this type of research, theoretical and experimental disciplines merge in order to find solutions to real-world problems. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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11 pages, 269 KiB

Open AccessFeature PaperArticle

Voronovskaja-Type Quantitative Results for Differences of Positive Linear Operators

by Ana Maria Acu, Gülen Başcanbaz-Tunca and Ioan Raşa

Symmetry 2021, 13(8), 1392; https://doi.org/10.3390/sym13081392 - 31 Jul 2021

Cited by 3 | Viewed by 1644

Abstract

We consider positive linear operators having the same fundamental functions and different functionals in front of them. For differences involving such operators, we obtain Voronovskaja-type quantitative results. Applications illustrating the theoretical aspects are presented. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

19 pages, 288 KiB

Open AccessArticle

General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces

by Silvestru Sever Dragomir

Symmetry 2021, 13(7), 1288; https://doi.org/10.3390/sym13071288 - 17 Jul 2021

Viewed by 1457

Abstract

In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

16 pages, 1362 KiB

Open AccessFeature PaperArticle

Some New Results Concerning the Classical Bernstein Cubature Formula

by Dan Miclăuş

Symmetry 2021, 13(6), 1068; https://doi.org/10.3390/sym13061068 - 15 Jun 2021

Viewed by 1786

Abstract

In this article, we present a solution to the approximation problem of the volume obtained by the integration of a bivariate function on any finite interval

[a, b] \times [c, d]

, as well as on any [...] Read more.

In this article, we present a solution to the approximation problem of the volume obtained by the integration of a bivariate function on any finite interval

[a, b] \times [c, d]

, as well as on any symmetrical finite interval

[- a, a] \times [- a, a]

when a double integral cannot be computed exactly. The approximation of various double integrals is done by cubature formulas. We propose a cubature formula constructed on the base of the classical bivariate Bernstein operator. As a valuable tool to approximate any volume resulted by integration of a bivariate function, we use the classical Bernstein cubature formula. Numerical examples are given to increase the validity of the theoretical aspects. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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24 pages, 346 KiB

Open AccessArticle

A Parametric Generalization of the Baskakov-Schurer-Szász-Stancu Approximation Operators

by Naim Latif Braha, Toufik Mansour and Hari Mohan Srivastava

Symmetry 2021, 13(6), 980; https://doi.org/10.3390/sym13060980 - 31 May 2021

Cited by 18 | Viewed by 2097

Abstract

In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem [...] Read more.

In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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