EditorialOperator Methods in Approximation Theory

Vita Leonessa ,1 Tuncer Acar ,2 Mirella Cappelletti Montano,3 and Pedro Garrancho 4

1Department of Mathematics, Computer Science and Economics, University of Basilicata, Potenza, Italy2Department of Mathematics, Faculty of Science, Selcuk University, Selcuklu, Konya, Turkey3Department of Mathematics, University of Bari, Bari, Italy4Department of Mathematics, University of Jaen, Jaen, Spain

Correspondence should be addressed to Vita Leonessa; vita.leonessa@unibas.it

Received 16 June 2019; Accepted 16 June 2019; Published 11 July 2019

Copyright © 2019 Vita Leonessa et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Approximation theory is an intensive research area, devel-oped in different directions by many mathematicians.

For example, approximation and iteration processes arisein a very natural way in many problems dealing with theconstructive approximation of functions as well as solutionsto (partial) differential equations and integral equations.Moreover, approximation theory can be successfully appliedin fixed point theory, in computer aided geometric design, inartificial neural networks, in the study of evolution problems,and in function algebras.

The goal of this special issue is to attract original researchand review articles that highlight recent advances in operatormethods within approximation theory and related applica-tions.

The interest aroused by the mathematicians who workin this area is remarkable, as evidenced by the thirty-sixsubmissions received.

The papers that have been accepted for the publication inthe issue recover the following topics:means and inequalities,approximation by positive operators, function algebras, fixedpoint theorems, and iteration processes.

In what follows we give a brief description of the contentsof this special issue.

In the review article titled “On Sequences of J. P. King-TypeOperators,” T. Acar et al. provide an essential expositionof a series of investigations developed in the last fifteenyears after the release of a paper written by J. P. King,where a modification of the classical Bernstein operators wasconsidered in order to get better approximation propertiesthan the original ones. After a brief history devoted to the

sequences of positive linear operators fixing certain (poly-nomial, exponential, or more general) functions obtained byapplying King’s approach, the authors illustrate certain King-type modifications of the well-known Bernstein and Szasz-Mirakjan operators.

A. A. Bakery and M. M. Mohammed in the paper “SmallPre-Quasi Banach Operator Ideals of Type Orlicz-CesaroMean Sequence Spaces” deal with determining sufficientconditions on an Orlicz-Cesaro mean sequence space 𝑐𝑒𝑠𝜑in order that the class 𝑆𝑐𝑒𝑠𝜑 , consisting of all bounded linearoperators between arbitrary Banach spaces such that thecorresponding sequence of 𝑠-numbers belongs to 𝑐𝑒𝑠𝜑, formsan operator ideal. Moreover, the authors determine someinclusion relations between pre-quasi operator ideals as wellas their duals. Finally, they present sufficient conditionson 𝑐𝑒𝑠𝜑 in order that the pre-quasi Banach operator idealgenerated by approximation number is small.

The purpose of the paper “Asymptotic Behavior ofAlmost Quartic ∗-Derivations on Banach ∗-Algebras” by H.-M. Kim et al. is determining, in the context of Banach ∗-algebras, stability theorems of quartic ∗-derivations associ-ated with the quartic functional equation f (3x − y) + f (x + y)+ 6f (x − y) = 4f (2x − y) + 4f (y) + 24f (x).

S. Z. Ullah et al. generalize and improve some knownresults concerning integral majorization type and generalizedFavard’s inequalities for the class of strongly convex functionsin the paper titled “IntegralMajorizationType Inequalities forthe Functions in the Sense of Strong Convexity.”

In the paper “Bivariate Chlodowsky-Stancu Variant of(p, q)-Bernstein-Schurer Operators” by T. Vedi-Dilek and E.

HindawiJournal of Function SpacesVolume 2019, Article ID 3426301, 2 pageshttps://doi.org/10.1155/2019/3426301

2 Journal of Function Spaces

Gemikonakli, the bivariate Chlodowsky-Stancu variant of(𝑝, 𝑞)-Bernstein-Schurer Operators, as well as a generaliza-tion of it, is proposed and some approximation properties areinvestigated. The paper ends by discussing some numericalresults.

N. Ozmen in the paper titled “New Generating Func-tion Relations for the q-Generalized Cesaro Polynomials”examines a 𝑞-analogue of generalized Cesaro polynomials forwhich she derives bilinear and bilateral generating functions;in addition, she gets a specific linear 𝑞-generating relationshipthat recovers the basic analogue of certain special functions.

H. J. Lee proves that the 𝑘-homogeneous polynomialand analytic numerical index of certain 𝑋-valued functionalgebras, 𝑋 being a complex Banach space, are the same asthose of 𝑋 in the paper titled “Generalized Numerical Indexof Function.”

In the paper “Convergence Analysis of an AcceleratedIteration for Monotone Generalized 𝛼-Nonexpansive Map-pings with a Partial Order,” a new accelerated iterationprocess for finding fixed points of monotone generalized 𝛼-nonexpansive mappings in ordered Banach spaces is intro-duced. Y.-A. Chen and D.-J. Wen establish some weak andstrong convergence theorems of fixed point for monotonegeneralized 𝛼-nonexpansive mappings in a uniformly convexpartially ordered Banach space. Moreover, they provide anumerical example that illustrates the convergence behaviorand effectiveness of their method.

C. Zhang and S. Wang in the paper titled “StructureProperties for Binomial Operators” discussed some struc-tures properties of the binomial operators, such as momentsrepresentation, derivatives representation, and binary rep-resentation. As applications, the authors prove that thebinomial operators considered preserve increasing func-tions, convex functions, and Holder (continuous) func-tions.

The purpose of the paper “OnNewPicard-Mann IterativeApproximations with Mixed Errors for Implicit MidpointRule and Applications” written by T. Li and H. Lan is tointroduce and study a new class of Picard-Mann iterationprocesses with mixed errors for the implicit midpoint rulesand to analyze the convergence and stability of the proposedmethod. Some numerical examples and applications to opti-mal control problemswith elliptic boundary value constraintsare presented, and they show that the Picard-Mann iterationprocess discussed in the article is more effective than otherrelated iterative processes.

J.-L. Wang et al. in the paper titled “On Approximatingthe Toader Mean by Other Bivariate Means” provide severalsharp bounds for the Toader mean by using certain com-binations of the arithmetic, quadratic, contraharmonic, andGaussian arithmetic geometric means.

The paper “𝐶∗ -Basic Construction from the ConditionalExpectation on the Drinfeld Double” by Q. Xin et al. dealswith the following topics. Let 𝐺 be a finite group and 𝐻a subgroup of 𝐺. Starting from the Drinfeld double 𝐷(𝐺)and the crossed product 𝐷(𝐺;𝐻) of 𝐶(𝐺) and C𝐻, andconsidering the 𝐶∗-basic construction 𝐶∗fbffkD(G), efbfftfrom the conditional expectation E of D(G) onto D(G; H),the authors construct a crossed product𝐶∗-algebra C(G/H ×

G) ⋊ CG, in such a way that 𝐶∗fbffkD(G), efbfft is𝐶∗-algebraisomorphic to C(G/H × G) ⋊ CG.

In the paper “On a New Stability Problem of Radicalnth-Degree Functional Equation by Brzdęk’s Fixed-PointMethod,” D. Kang and H. B. Kim, given a positive integern, discuss the general solutions to the radical n-th degreefunctional equation f ( 𝑛√𝑥𝑛 + 𝑦𝑛) = f (x) + f (y) and prove newHyers-Ulam type stability results by using Brzdek’s fixed pointmethod.

In the context of singular Hadamard fractional boundaryvalue problems, J. Mao et al. establish, by using an iterativealgorithm, the existence and uniqueness of the exact iterativesolution in the paper titled “The Unique Positive Solution forSingular Hadamard Fractional Boundary Value Problems.”Moreover, they show the iterative sequences converge uni-formly to the exact solution, and they provide estimation ofthe approximation error and the convergence rate.

Conflicts of Interest

The Guest Editors declare that they have no conflicts ofinterest regarding the publication of this special issue.

Acknowledgments

We are very grateful to the mathematical community forthe great interest shown in this special issue. In particular,we want to thank all the authors of the published papersfor their contribution in this field. Also, our gratitude goesto the reviewers, for their precious help in handling all thesubmitted manuscripts, and to Hindawi who supported usthroughout the development of this special issue. Finally, theLead Guest Editor sincerely thanks her Guest Editors foragreeing to join her in this project.

Vita LeonessaTuncer Acar

Mirella Cappelletti MontanoPedro Garrancho

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