Approximation Theory, Sequence Spaces and Applications

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. Preface 6 Contents 9 About the Editors 11 1 Topology on Geometric Sequence Spaces 13 1.1 Introduction 13 1.1.1 α-Generator and Geometric Complex Field 14 1.1.2 Some Useful Relations Between Geometric Operations and Ordinary Arithmetic Operations 16 1.1.3 G-Limit 18 1.1.4 G-Continuity 19 1.2 Geometric Vector Spaces 19 1.2.1 Geometric Vector Space 19 1.2.2 Dual System 20 1.3 Topology on Geometric Sequence Spaces 22 1.3.1 Normal Topology 24 1.3.2 Perfect Sequence Space 26 1.3.3 Simple Space 28 1.3.4 Symmetric Sequence Spaces 29 References 30 2 Composition Operators on Second-Order Cesàro Function Spaces 32 2.1 Introduction 32 2.2 Examining the Boundedness 35 2.3 Compactness and Essential Norm of Composition Operators 37 2.4 Fredholm Composition Operators 40 2.5 Conclusion 43 References 43 3 Generalized Deferred Statistical Convergence 45 3.1 Definitions and Preliminaries 46 3.2 Deferred Statistical Convergence of Order αβ 49 3.3 Strong s-Deferred Cesàro Summability of Order αβ 52 3.4 Inclusion Theorems 55 3.5 Special Cases 57 References 61 4 Approximation by Generalized Lupaş-Pǎltǎnea Operators 63 4.1 Introduction 63 4.2 Basic Results 65 4.3 Main Results 66 4.3.1 Weighted Approximation 67 4.3.2 Quantitative Voronoskaja-Type Approximation Theorem 68 4.3.3 Grüss Voronovskaya-Type Theorem 71 4.3.4 Approximation Properties of DBV[0,infty) 71 References 78 5 Zachary Spaces mathcalZp[mathbbRinfty] and Separable Banach Spaces 80 5.1 Introduction 80 5.1.1 Preliminaries 81 5.1.2 Basis for a Banach Spaces 82 5.2 Space of Functions of Bounded Mean Oscillation (BMO[mathbbRIinfty]) 83 5.3 Zachary Space mathcalZp[mathbbRIinfty] 86 5.4 Zachary Space mathcalZp[mathfrakB], Where mathfrakB is Separable Banach Space 88 References 90 6 New Generalization of the Power Summability Methods for Dunkl Generalization of Szász Operators via q-Calculus 92 6.1 Introduction 93 6.2 Dunkl Generalization of the Szász Operators Obtained by q-Calculus 94 6.3 Preliminary Results 95 6.4 Direct Estimates 96 6.5 Weighted Approximation 104 6.6 Statistical Approximation Properties for Dunkl Generalization of Szász Operators via q-Calculus 109 6.7 Rate of Convergence of the Dunkl Generalization of Szász Operators via q-Calculus 115 6.8 Conclusion 124 References 124 7 Approximation by Generalized Szász–Jakimovski–Leviatan Type Operators 127 7.1 Introduction 127 7.2 Construction of Operators and Estimation of Moments 129 7.3 Approximation in Weighted Spaces 133 7.4 Some Direct Approximation Theorems 136 7.5 A-Statistical Convergence 140 7.6 Conclusion 143 References 143 8 On Approximation of Signals 146 8.1 Introduction 146 8.2 Known Results 151 8.3 Main Theorems 152 8.4 Lemmas 153 8.5 Proof of the Lemmas 155 8.6 Proof of Main Theorems 158 8.7 Conclusion 167 References 167 9 Numerical Solution for Nonlinear Problems 170 9.1 Introduction 170 9.2 Introducing Some Nonlinear Functional and Fractional Equations 171 9.3 A Coupled Semi-analytic Method to Find the Solution of Equation (9.1) 173 9.3.1 Constructing Some Iterative Algorithms to Approximate the Solution of Equations (9.2)–(9.5) 174 9.4 Convergence of the Algorithms 184 9.5 Constructing an Iterative Algorithm by Sinc Function 186 9.5.1 One-Dimensional Functional Integral Equation 188 9.5.2 Convergence of Algorithm (9.62) 189 9.5.3 Two-Dimensional Functional Integral Equation 189 References 191 10 Szász-Type Operators Involving q-Appell Polynomials 194 10.1 Introduction 194 10.2 Construction of the Operators and Basic Estimates 196 10.3 Some Basic Results 197 10.4 Pointwise Approximation Results 201 10.5 Weighted Approximation 204 10.6 A-Statistical Approximation 207 References 208 11 Commutants of the Infinite Hilbert Operators 210 11.1 Introduction 210 11.2 Main Results 214 11.3 Norm of Operators on Sequence Spaces Φn(p) and Ψn(p) 220 References 225 12 On Complex Uncertain Sequences Defined by Orlicz Function 227 12.1 Introduction 227 12.2 Preliminaries 228 12.3 Complex Uncertain Sequence Spaces 230 12.4 Statistical Convergence of Complex Uncertain Sequences 232 12.5 Complex Uncertain Sequence Spaces Defined by Orlicz Function 235 12.6 Statistical Convergence of Complex Uncertain Sequences Defined by Orlicz Function 237 12.7 On Paranormed Type p-Absolutely Summable Uncertain Sequence Spaces Defined by Orlicz Functions 239 12.8 Lacunary Convergence Concepts of Complex Uncertain Sequences with Respect to Orlicz Function 242 12.9 Conclusion 245 References 245 13 Ulam-Hyers Stability of Mixed Type Functional Equation Deriving From Additive and Quadratic Mappings in Intuitionistic Random Normed Spaces 248 13.1 Introduction 249 13.2 Preliminaries 250 13.3 Ulam-Hyers Stability for Odd Case 252 13.4 Ulam-Hyers Stability for Even Case 254 13.5 Ulam-Hyers Stability for Mixed Case 257 13.6 Conclusion 259 References 259 14 A Study on q-Euler Difference Sequence Spaces 261 14.1 Introduction, Preliminaries, and Notations 261 14.1.1 Euler Matrix of Order 1 and Sequence Spaces 262 14.1.2 q-Calculus 263 14.2 q-Euler Difference Sequence Spaces 265 14.3 Alpha-, Beta-, and Gamma-Duals of q-Euler Difference Sequence Spaces 267 14.4 Matrix Transformations 269 14.5 Compact Operators and Hausdorff Measure of Non-compactness (Hmnc) 272 References 276