New Directions in Theory of Approximation and Related Problems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 September 2022) | Viewed by 16680

Special Issue Editors

Associate Professor, Department of Mathematics, Informatics, “Vasile Alecsandri” University of Bacău, 600115 Bacău, Romania
Interests: approximation theory using linear and positive operators; probability and statistics; numerical methods
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, 550012 Sibiu, Romania
Interests: approximation theory; numerical analysis; probability and statistics
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, Technical University of Cluj Napoca, 400114 Cluj-Napoca, Romania
Interests: approximation by positive linear operators; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Department of Statistics, Forecasting and Mathematics, Faculty of Economics and Business Administration, Babes Bolyai University of Cluj-Napoca, 400591 Cluj Napoca, Romania
Interests: approximation theory; linear positive operators
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to collect original and significant contributions dealing with both the theory and applications of approximation theory and related topics. In recent decades, with the development of computational techniques but also of generating increasingly complex problems, the theory of approximation has known new directions of research and development.

Thus, this Special Issue provides new trends in the field of approximation theory and related applications, a field which is an important bridge between pure and applied mathematics.

The present issue covers topics in classical approximation, approximation using linear and positive operators, degree of approximation, interpolation and quadratures, multivariate approximation, spline functions of one and several variables, numerical algorithms, numerical methods for partial differential equations, and fixed point theory and its applications.

Prof. Dr. Carmen Violeta Muraru
Prof. Dr. Ana-Maria Acu
Prof. Dr. Marius Birou
Dr. Radu Voichita Adriana
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • approximation theory
  • degree of approximation
  • numerical methods
  • partial differential equations
  • fixed point

Published Papers (10 papers)

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Research

15 pages, 2256 KiB  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym14040662 - 24 Mar 2022
Cited by 3 | Viewed by 1453
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  
Article
Yet Another New Variant of Szász–Mirakyan Operator
by Ana Maria Acu and Gancho Tachev
Symmetry 2021, 13(11), 2018; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112018 - 25 Oct 2021
Cited by 7 | Viewed by 1622
Abstract
In this paper, we construct a new variant of the classical Szász–Mirakyan operators, Mn, which fixes the functions 1 and eax,x0,aR. 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, Mn, which fixes the functions 1 and eax,x0,aR. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of Mn 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  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym13101919 - 12 Oct 2021
Cited by 19 | Viewed by 1576
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  
Article
On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators
by Ulrich Abel and Octavian Agratini
Symmetry 2021, 13(10), 1841; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101841 - 01 Oct 2021
Viewed by 1255
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  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym13091747 - 19 Sep 2021
Viewed by 1264
Abstract
In the present paper, we propose a Baskakov operator of integral type using a function φ on [0,) with the properties: φ(0)=0,φ>0 on [0,) and [...] Read more.
In the present paper, we propose a Baskakov operator of integral type using a function φ on [0,) with the properties: φ(0)=0,φ>0 on [0,) and limxφ(x)=. 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 (,0] (the symmetric of the interval [0,) 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  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym13081450 - 09 Aug 2021
Cited by 1 | Viewed by 1432
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  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym13081392 - 31 Jul 2021
Cited by 3 | Viewed by 1300
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  
Article
General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces
by Silvestru Sever Dragomir
Symmetry 2021, 13(7), 1288; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071288 - 17 Jul 2021
Viewed by 1137
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  
Article
Some New Results Concerning the Classical Bernstein Cubature Formula
by Dan Miclăuş
Symmetry 2021, 13(6), 1068; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061068 - 15 Jun 2021
Viewed by 1391
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]×[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]×[c,d], as well as on any symmetrical finite interval [a,a]×[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  
Article
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://0-doi-org.brum.beds.ac.uk/10.3390/sym13060980 - 31 May 2021
Cited by 15 | Viewed by 1699
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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