This book publishes original research chapters on the theory of approximation by positive linear operators as well as theory of sequence spaces and illustrates their applications. Chapters are original and contributed by active researchers in the field of approximation theory and sequence spaces. Each chapter describes the problem of current importance and summarizes ways of their solution and possible applications which improve the current understanding pertaining to sequence spaces and approximation theory. The presentation of the articles is clear and self-contained throughout the book.
Author(s): S. A. Mohiuddine, Bipan Hazarika, Hemant Kumar Nashine
Series: Industrial and Applied Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 276
City: Singapore
Preface
Contents
About the Editors
1 Topology on Geometric Sequence Spaces
1.1 Introduction
1.1.1 α-Generator and Geometric Complex Field
1.1.2 Some Useful Relations Between Geometric Operations and Ordinary Arithmetic Operations
1.1.3 G-Limit
1.1.4 G-Continuity
1.2 Geometric Vector Spaces
1.2.1 Geometric Vector Space
1.2.2 Dual System
1.3 Topology on Geometric Sequence Spaces
1.3.1 Normal Topology
1.3.2 Perfect Sequence Space
1.3.3 Simple Space
1.3.4 Symmetric Sequence Spaces
References
2 Composition Operators on Second-Order Cesàro Function Spaces
2.1 Introduction
2.2 Examining the Boundedness
2.3 Compactness and Essential Norm of Composition Operators
2.4 Fredholm Composition Operators
2.5 Conclusion
References
3 Generalized Deferred Statistical Convergence
3.1 Definitions and Preliminaries
3.2 Deferred Statistical Convergence of Order αβ
3.3 Strong s-Deferred Cesàro Summability of Order αβ
3.4 Inclusion Theorems
3.5 Special Cases
References
4 Approximation by Generalized Lupaş-Pǎltǎnea Operators
4.1 Introduction
4.2 Basic Results
4.3 Main Results
4.3.1 Weighted Approximation
4.3.2 Quantitative Voronoskaja-Type Approximation Theorem
4.3.3 Grüss Voronovskaya-Type Theorem
4.3.4 Approximation Properties of DBV[0,infty)
References
5 Zachary Spaces mathcalZp[mathbbRinfty] and Separable Banach Spaces
5.1 Introduction
5.1.1 Preliminaries
5.1.2 Basis for a Banach Spaces
5.2 Space of Functions of Bounded Mean Oscillation (BMO[mathbbRIinfty])
5.3 Zachary Space mathcalZp[mathbbRIinfty]
5.4 Zachary Space mathcalZp[mathfrakB], Where mathfrakB is Separable Banach Space
References
6 New Generalization of the Power Summability Methods for Dunkl Generalization of Szász Operators via q-Calculus
6.1 Introduction
6.2 Dunkl Generalization of the Szász Operators Obtained by q-Calculus
6.3 Preliminary Results
6.4 Direct Estimates
6.5 Weighted Approximation
6.6 Statistical Approximation Properties for Dunkl Generalization of Szász Operators via q-Calculus
6.7 Rate of Convergence of the Dunkl Generalization of Szász Operators via q-Calculus
6.8 Conclusion
References
7 Approximation by Generalized Szász–Jakimovski–Leviatan Type Operators
7.1 Introduction
7.2 Construction of Operators and Estimation of Moments
7.3 Approximation in Weighted Spaces
7.4 Some Direct Approximation Theorems
7.5 A-Statistical Convergence
7.6 Conclusion
References
8 On Approximation of Signals
8.1 Introduction
8.2 Known Results
8.3 Main Theorems
8.4 Lemmas
8.5 Proof of the Lemmas
8.6 Proof of Main Theorems
8.7 Conclusion
References
9 Numerical Solution for Nonlinear Problems
9.1 Introduction
9.2 Introducing Some Nonlinear Functional and Fractional Equations
9.3 A Coupled Semi-analytic Method to Find the Solution of Equation (9.1)
9.3.1 Constructing Some Iterative Algorithms to Approximate the Solution of Equations (9.2)–(9.5)
9.4 Convergence of the Algorithms
9.5 Constructing an Iterative Algorithm by Sinc Function
9.5.1 One-Dimensional Functional Integral Equation
9.5.2 Convergence of Algorithm (9.62)
9.5.3 Two-Dimensional Functional Integral Equation
References
10 Szász-Type Operators Involving q-Appell Polynomials
10.1 Introduction
10.2 Construction of the Operators and Basic Estimates
10.3 Some Basic Results
10.4 Pointwise Approximation Results
10.5 Weighted Approximation
10.6 A-Statistical Approximation
References
11 Commutants of the Infinite Hilbert Operators
11.1 Introduction
11.2 Main Results
11.3 Norm of Operators on Sequence Spaces Φn(p) and Ψn(p)
References
12 On Complex Uncertain Sequences Defined by Orlicz Function
12.1 Introduction
12.2 Preliminaries
12.3 Complex Uncertain Sequence Spaces
12.4 Statistical Convergence of Complex Uncertain Sequences
12.5 Complex Uncertain Sequence Spaces Defined by Orlicz Function
12.6 Statistical Convergence of Complex Uncertain Sequences Defined by Orlicz Function
12.7 On Paranormed Type p-Absolutely Summable Uncertain Sequence Spaces Defined by Orlicz Functions
12.8 Lacunary Convergence Concepts of Complex Uncertain Sequences with Respect to Orlicz Function
12.9 Conclusion
References
13 Ulam-Hyers Stability of Mixed Type Functional Equation Deriving From Additive and Quadratic Mappings in Intuitionistic Random Normed Spaces
13.1 Introduction
13.2 Preliminaries
13.3 Ulam-Hyers Stability for Odd Case
13.4 Ulam-Hyers Stability for Even Case
13.5 Ulam-Hyers Stability for Mixed Case
13.6 Conclusion
References
14 A Study on q-Euler Difference Sequence Spaces
14.1 Introduction, Preliminaries, and Notations
14.1.1 Euler Matrix of Order 1 and Sequence Spaces
14.1.2 q-Calculus
14.2 q-Euler Difference Sequence Spaces
14.3 Alpha-, Beta-, and Gamma-Duals of q-Euler Difference Sequence Spaces
14.4 Matrix Transformations
14.5 Compact Operators and Hausdorff Measure of Non-compactness (Hmnc)
References
