Mathematical Analysis, Approximation Theory and Their Applications

Frontmatter

A Brief History of the Favard Operator and Its Variants

Abstract

In the year 1944 the well-known French mathematician Jean Favard (1902–1965) introduced a discretely defined operator which is a discrete analogue of the familiar Gauss–Weierstrass singular convolution integral. In the present chapter we sketch the history of this approximation operator during the past 70 years by presenting known results on the operator and its various extensions and variants. The first part after the introduction is dedicated to saturation of the classical Favard operator in weighted Banach spaces. Furthermore, we discuss the asymptotic behaviour of a slight generalization \(F_{n,\sigma _{n}}\) of the Favard operator and its Durrmeyer variant \(\tilde{F }_{n,\sigma _{n}}\). In particular, the local rate of convergence when applied to locally smooth functions is considered. The main result of this part consists of the complete asymptotic expansions for the sequences \(\left (F_{n,\sigma _{n}}f\right )\left (x\right )\) and \(\left (\tilde{F } _{n,\sigma _{n}}f\right )\left (x\right )\) as n tends to infinity. Furthermore, these asymptotic expansions are valid also with respect to simultaneous approximation. A further part is devoted to the recent work of several Polish mathematicians on approximation in weighted function spaces. Finally, we define left quasi-interpolants for the Favard operator and its Durrmeyer variant in the sense of Sablonnière.

Ulrich Abel

Bivariate Extension of Linear Positive Operators

Abstract

The goal of this chapter is to present a survey of the literature on approximation of functions of two variables by linear positive operators. We study the approximation properties of these operators in the space of functions of two variables, continuous on a compact set. We also discuss the convergence of the operators in a weighted space of functions of two variables and find the rate of this convergence by means of modulus of continuity.

P. N. Agrawal, Meenu Goyal

Positive Green’s Functions for Boundary Value Problems with Conformable Derivatives

Abstract

We use a newly introduced conformable derivative to formulate several boundary value problems with three or four conformable derivatives, including those with conjugate, right-focal, and Lidstone conditions. With the conformable differential equation and boundary conditions established, we find the corresponding Green’s functions and prove their positivity under appropriate assumptions.

Douglas R. Anderson

The Retraction-Displacement Condition in the Theory of Fixed Point Equation with a Convergent Iterative Algorithm

Abstract

Let (X, d) be a complete metric space and f: X → X be an operator with a nonempty fixed point set, i.e., \(F_{f}:=\{ x \in X: x = f(x)\}\neq \emptyset\). We consider an iterative algorithm with the following properties:

(1)

for each x ∈ X there exists a convergent sequence (x _n (x)) such that \(x_{n}(x) \rightarrow x^{{\ast}}(x) \in F_{f}\) as \(n \rightarrow \infty\);

 

(2)

if x ∈ F _f , then x _n (x) = x, for all \(n \in \mathbb{N}\).

 

In this way, we get a retraction mapping r: X → F _f , given by r(x) = x ^∗(x). Notice that, in the case of Picard iteration, this retraction is the operator \(f^{\infty }\), see I.A. Rus (Picard operators and applications, Sci. Math. Jpn. 58(1):191–219, 2003).By definition, the operator f satisfies the retraction-displacement condition if there is an increasing function \(\psi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) which is continuous at 0 and satisfies ψ(0) = 0, such that

$$\displaystyle{d(x,r(x)) \leq \psi (d(x,f(x)),\mbox{ for all }x \in X.}$$

In this paper, we study the fixed point equation x = f(x) in terms of a retraction-displacement condition. Some examples, corresponding to Picard, Krasnoselskii, Mann and Halpern iterative algorithms, are given. Some new research directions and open questions are also presented.

V. Berinde, A. Petruşel, I. A. Rus, M. A. Şerban

An Adaptive Finite Element Method for Solving a Free Boundary Problem with Periodic Boundary Conditions in Lubrication Theory

Abstract

A duality approximation method combined with an adaptive finite element method is applied to solve a free boundary problem with periodic boundary conditions in lubrication theory.

O. Chau, D. Goeleven, R. Oujja

Evolution Solutions of Equilibrium Problems: A Computational Approach

Abstract

This paper proposes a computational method to describe evolution solutions of known classes of time-dependent equilibrium problems (such as time-dependent traffic network, market equilibrium or oligopoly problems, and dynamic noncooperative games). Equilibrium solutions for these classes have been studied extensively from both a theoretical (regularity, stability behaviour) and a computational point of view. In this paper we highlight a method to further study the solution set of such problems from a dynamical systems perspective, namely we study their behaviour when they are not in an (market, traffic, financial, etc.) equilibrium state. To this end, we define what is meant by an evolution solution for a time-dependent equilibrium problem and we introduce a computational method for tracking and visualizing evolution solutions using a projected dynamical system defined on a carefully chosen L ²-space. We strengthen our results with various examples.

Monica-Gabriela Cojocaru, Scott Greenhalgh

Cantor, Banach and Baire Theorems in Generalized Metric Spaces

Abstract

In the paper we present basic results for generalized metric spaces: Cantor, Banach and Baire theorems, well known and widely applied in metric spaces and mathematics.

Stefan Czerwik, Krzysztof Król

A Survey of Perturbed Ostrowski Type Inequalities

Abstract

In this paper we survey a number of recent perturbed versions of Ostrowski inequality that have been obtained by the author and provide their connections with numerous classical results of interest.

Silvestru Sever Dragomir

Hyers–Ulam–Rassias Stability of the Generalized Wilson’s Functional Equation

Abstract

In this chapter, we apply the fixed point theorem and the direct method to the proof of Hyers–Ulam–Rassias stability property for generalized Wilson’s functional equation

$$\displaystyle\begin{array}{rcl} \int _{K}\int _{G}f(xtk.y)dkd\mu (t) = f(x)g(y),\;x,y \in G,& & {}\\ \end{array}$$

where f, g are continuous complex valued functions on a locally compact group G, K is a compact subgroup of morphisms of G, dk is the normalized Haar measure on K and μ is a K-invariant complex measure with compact support.

Elqorachi Elhoucien, Youssef Manar, Sammad Khalil

Approximation Under Exponential Growth Conditions by Szász and Baskakov Type Operators in the Complex Plane

Abstract

In this chapter, firstly for q > 1 the exact order near to the best approximation, \(\frac{1} {q^{n}}\), is obtained in approximation by complex q-Favard–Szász–Mirakjan, q-Szász–Kantorovich operators and q-Baskakov operators attached to functions of exponential growth conditions, which are entire functions or analytic functions defined only in compact disks (without to require to be defined on the whole axis \([0,+\infty )\)). Quantitative Voronovskaja-type results of approximation order \(\frac{1} {q^{2n}}\) are proved. For q-Szász–Kantorovich operators, the case q = 1 also is considered, when the exact order of approximation \(\frac{1} {n}\) is obtained. Approximation results for a link operator between the Phillips and Favard–Szász–Mirakjan operators are also obtained. Then, by using a sequence \(\frac{b_{n}} {a_{n}}:=\lambda _{n}> 0\), \(a_{n},b_{n}> 0\), \(n \in \mathbb{N}\) with the property that \(\lambda _{n} \rightarrow 0\) as fast we want, we obtain the approximation order \(O(\lambda _{n})\) for the generalized Szász–Faber operators and the generalized Baskakov–Faber operators attached to analytic functions of exponential growth in a continuum \(G \subset \mathbb{C}\). Several concrete examples of continuums G are given for which these generalized operators can explicitly be constructed. Finally, approximation results for complex Baskakov–Szász–Durrmeyer operators are presented.

Sorin G. Gal

On the Asymptotic Behavior of Sequences of Positive Linear Approximation Operators

Abstract

We provide an analysis of the rate of convergence of positive linear approximation operators defined on C[0, 1]. We obtain a sufficient condition for a sequence of positive linear approximation operators to possess a Mamedov-type property and give an application to the Durrmeyer approximation process.

Ioan Gavrea, Mircea Ivan

Approximation of Functions by Additive and by Quadratic Mappings

Abstract

In this chapter, we characterize the functions with values in a Banach space which can be approximated by additive mappings, with a given error. Also, we give a characterization of functions with values in a Banach space which can be approximated by a quadratic mapping, with a given error.

Laura Găvruţa, Paşc Găvruţa

Bernstein Type Inequalities Concerning Growth of Polynomials

Abstract

Let \(p(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + \cdots + a_{n}z^{n}\) be a polynomial of degree n, where the coefficients a _j , for 0 ≤ j ≤ n, may be complex, and p(z) ≠ 0 for | z | < 1. Then

$$\displaystyle{ M(p,R) \leq \Big (\frac{R^{n} + 1} {2} \Big)\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ R \geq 1, }$$

(1)

and

$$\displaystyle{ M(p,r) \geq \Big (\frac{r + 1} {2} \Big)^{n}\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ 0 <r \leq 1, }$$

(2)

where \(M(p,R):= \max _{\vert z\vert =R\geq 1}\vert p(z)\vert\), \(M(p,r):= \max _{\vert z\vert =r\leq 1}\vert p(z)\vert\), and \(\vert \vert p\vert \vert:= \max _{\vert z\vert =1}\vert p(z)\vert\). Inequality (1) is due to Ankeny and Rivlin (Pac. J. Math. 5, 849–852, 1955), whereas Inequality (2) is due to Rivlin (Am. Math. Mon. 67, 251–253, 1960). These inequalities, which due to their applications are of great importance, have been the starting point of a considerable literature in Approximation Theory, and in this paper we study some of the developments that have taken place around these inequalities. The paper is expository in nature and would provide results dealing with extensions, generalizations and refinements of these inequalities starting from the beginning of this subject to some of the recent ones.

N. K. Govil, Eze R. Nwaeze

Approximation for Generalization of Baskakov–Durrmeyer Operators

Abstract

In the present article, we study certain approximation properties of the modified form of generalized Baskakov operators introduced by Erencin (Appl. Math. Comput. 218(3):4384–4390, 2011). We estimate a recurrence relation for the moments of their Durrmeyer type modification. First we estimate rate of convergence for functions having derivatives of bounded variation. Next, we discuss some direct results in simultaneous approximation by these operators, e.g. point-wise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity.

Vijay Gupta

A Tour on p(x)-Laplacian Problems When p = ∞

Abstract

Most of the times, in problems where the p(x)-Laplacian is involved, the variable exponent p(⋅ ) is assumed to be bounded. The main reason for this is to be able to apply standard variational methods. The aim of this paper is to present the work that has been done so far, in problems where the variable exponent p(⋅ ) equals infinity in some part of the domain. In this case the infinity Laplace operator arises naturally and the notion of weak solution does not apply in the part where p(⋅ ) becomes infinite. Thus the notion of viscosity solution enters into the picture. We study both the Dirichlet and the Neumann case.

Yiannis Karagiorgos, Nikos Yannakakis

An Umbral Calculus Approach to Bernoulli–Padé Polynomials

Abstract

In this paper, we consider Bernoulli–Padé polynomials of fixed order whose generating function is based on the Padé approximant of the exponential function. We derive, by using umbral calculus techniques, several recurrence relations for these polynomials and investigate connections between our polynomials and several known families of polynomials.

Dae San Kim, Taekyun Kim

Hadamard Matrices: Insights into Their Growth Factor and Determinant Computations

Abstract

In this expository paper we survey the most important progress in the growth problem for Hadamard matrices. The history of the problem is presented, the importance of determinant calculations is highlighted, and the relevant open problems are discussed. Emphasis is laid on the contribution of determinant manipulations to the study of the growth factor for Hadamard matrices after application of Gaussian Elimination with complete pivoting on them, which is an important scientific field in Numerical Analysis.

Christos D. Kravvaritis

Localized Summability Kernels for Jacobi Expansions

Abstract

While the direct and converse theorems of approximation theory enable us to characterize the smoothness of a function \(f: [-1,1] \rightarrow \mathbb{R}\) in terms of its degree of polynomial approximation, they do not account for local smoothness. The use of localized summability kernels leads to a wavelet-like representation, using the Fourier–Jacobi coefficients of f, so as to characterize the smoothness of f in a neighborhood of each point in terms of the behavior of the terms of this representation. In this paper, we study the localization properties of a class of kernels, which have explicit forms in the “space domain,” and establish explicit bounds on the Lebesgue constants on the summability kernels corresponding to some of these.

H. N. Mhaskar

Quadrature Rules with Multiple Nodes

Abstract

In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generalized quadrature rules considered by Ghizzeti and Ossicini (Quadrature Formulae, Academie, Berlin, 1970) and apply their ideas in order to obtain quadrature rules with multiple nodes and the maximal trigonometric degree of exactness. Such quadrature rules are characterized by the so-called s- and \(\sigma\)-orthogonal trigonometric polynomials. Numerical method for constructing such quadrature rules is given, as well as a numerical example to illustrate the obtained theoretical results.

Gradimir V. Milovanović, Marija P. Stanić

-Summability of Sequences of Linear Conservative Operators

Abstract

This work deals with the approximation of functions by sequences of linear operators. Here the classical convergence is replaced by matrix summability. Beyond the usual positivity of the operators involved in the approximation processes, more general conservative approximation properties are considered. Quantitative results, as well as results on asymptotic formulae and saturation are stated. It is the intention of the authors to show the way in which some concepts of generalized convergence entered Korovkin-type approximation theory. This is a survey work that gathers and orders the results stated by the authors and other researchers within the aforesaid subject.

Daniel Cárdenas-Morales, Pedro Garrancho

Simultaneous Weighted Approximation with Multivariate Baskakov–Schurer Operators

Abstract

We study the properties of weighted simultaneous approximation of multivariate Baskakov–Schurer operators. We obtain quantitative estimates with explicit constants of the weighted approximation error for the partial derivatives. Moreover, we analyze the behavior of the operators with respect to weighted Lipschitz functions. For this purpose, we first compute the best constants, \(M \in \mathbb{R}\), in the inequalities of the type \(A_{n,p}\left ((1 + \left \vert t\right \vert )^{r}\right ) \leq M(1 + \left \vert t\right \vert )^{r}\).

Antonio-Jesús López-Moreno, Joaquı́n Jódar-Reyes, José-Manuel Latorre-Palacios

Approximation of Discontinuous Functions by q-Bernstein Polynomials

Abstract

This chapter presents an overview of the results related to the q-Bernstein polynomials with q > 1 attached to discontinuous functions on [0, 1]. It is emphasized that the singularities of such functions located on the set

$$\displaystyle{\mathbb{J}_{q}:=\{ 0\} \cup \{ q^{-l}\}_{ l=0}^{\infty },\;\;q> 1,}$$

are definitive for the investigation of the convergence properties of their q-Bernstein polynomials.

Sofia Ostrovska, Ahmet Yaşar Özban

Nests, and Their Role in the Orderability Problem

Abstract

This chapter is divided into two parts. The first part is a survey of some recent results on nests and the orderability problem. The second part consists of results, partial results and open questions, all viewed in the light of nests. From connected LOTS, to products of LOTS and function spaces, up to the order relation in the Fermat Real Line.

Kyriakos Papadopoulos

Resolvent Operators for Some Classes of Integro-Differential Equations

Abstract

Explicit representations are constructed for the resolvents of the operators of the form \(B =\widehat{ A} + Q_{1}\) and \(\mathbf{B} =\widehat{ A}^{2} + Q_{2}\), where \(\widehat{A}\) and \(\widehat{A}^{2}\) are linear closed operators with known resolvents and Q ₁ and Q ₂ are perturbation operators embedding inner products of \(\widehat{A}\) and \(\widehat{A}^{2}\) as they appear in integro-differential equations and other applications.

I. N. Parasidis, E. Providas

Component Matrices of a Square Matrix and Their Properties

Abstract

The definition of the component matrices of a square matrix A is well-known [Lancaster]. This paper is concerned with all the basic properties of component matrices of a square matrix A, where \(A\epsilon M_{\nu \times \nu }(K),K = \mathbb{R}\), or \(K = \mathbb{C}\). This is very useful for the studies of the spectral resolution of a matrix function f(A), the convergence of sequences and series of matrices and also the convergence of matrix functions. It is also useful to solve differential equations and control system problems.

Dorothea Petraki, Nikolaos Samaras

Solutions of Some Types of Differential Equations and of Their Associated Physical Problems by Means of Inverse Differential Operators

Abstract

We present an operational method, involving an inverse derivative operator, in order to obtain solutions for differential equations, which describe a broad range of physical problems. Inverse differential operators are proposed to solve a variety of differential equations. Integral transforms and the operational exponent are used to obtain the solutions. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. Examples of solutions of physical problems, related to the mass, the heat and other processes of propagation are demonstrated by the developed operational technique. In particular, the evolutional type problems, the generalizations of the Black–Scholes, of the heat, of the Fokker–Plank and of the telegraph equations are considered as well as equations, involving the Laguerre derivative operator.

H. M. Srivastava, K. V. Zhukovsky

A Modified Pointwise Estimate on Simultaneous Approximation by Bernstein Polynomials

Abstract

We prove pointwise estimate for the approximation of the k-th derivative of the function f by the k-th derivative of its Bernstein operator B _n . We compare our result with similar estimates obtained earlier.

Gancho Tachev

Structural Fixed Point Results in Metric Spaces

Abstract

In Part 1, a class of anticipative contractions over quasi-ordered metric spaces is introduced and a corresponding lot of metrical fixed point theorems is formulated. The obtained facts include some well-known statements in the area due to Boyd and Wong or Matkowski, as well as a recent contribution due to Choudhury and Kundu (Demonstr Math 46:327–334, 2013). Further, in Part 2, a relative type version is given for the fixed point result in Leader (Math Jpn 24:17–24, 1979). Finally, in Part 3, an almost metric version is established for the 2008 Jachymski fixed point result (Proc Am Math Soc 136:1359–1373, 2008) involving Banach contractions over metric spaces endowed with a graph.

Mihai Turinici

Models of Fuzzy Linear Regression: An Application in Engineering

Abstract

The classical Linear Regression is an approximation for the creation of a model which connects a dependent variable y with one or more independent variables X, and it is subjected to some assumptions. The violation of these assumptions can influence negatively the power of the use of statistical regression and its quality. Nowadays, to exceed this problem, a new method has been introduced, and is in use, that is called fuzzy regression. Fuzzy regression is considered to be possibilistic, with the distribution function of the possibility to be connected with the membership function of fuzzy numbers. In this article, we examine three models of fuzzy regression, for confidence level h=0, for the case of crisp input values, fuzzy output values and fuzzy regression parameters, with an application to Hydrology, in two rainfall stations in Northern Greece.

Christos Tzimopoulos, Kyriakos Papadopoulos, Basil K. Papadopoulos

Properties of Functions of Generalized Bounded Variations

Abstract

Looking to the features of functions of bounded variation, the notion of bounded variation is generalized in many ways and different classes of functions of generalized bounded variations are introduced. In the present chapter introducing different classes of functions of generalized bounded variations their main interesting properties are discussed. Inter-relations between them and the classes related to them are given in the next section. Finally we try to present overall picture of Fourier analysis of these classes.

Rajendra G. Vyas