Higher Order Derivatives / Edition 1

by Satya Mukhopadhyay

Overview

The concept of higher order derivatives is useful in many branches of mathematics and its applications. As they are useful in many places, n^th order derivatives are often defined directly. Higher Order Derivatives discusses these derivatives, their uses, and the relations among them. It covers higher order generalized derivatives, including the Peano, d.l.V.P., and Abel derivatives; along with the symmetric and unsymmetric Riemann, Cesàro, Borel, L^P-, and Laplace derivatives.

Although much work has been done on the Peano and de la Vallée Poussin derivatives, there is a large amount of work to be done on the other higher order derivatives as their properties remain often virtually unexplored. This book introduces newcomers interested in the field of higher order derivatives to the present state of knowledge. Basic advanced real analysis is the only required background, and, although the special Denjoy integral has been used, knowledge of the Lebesgue integral should suffice.

Product Details

ISBN-13:	9780367381745
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Monographs and Surveys in Pure and Applied Mathematics , #122
Pages:	220
Product dimensions:	6.12(w) x 9.19(h) x (d)

Table of Contents

Preface xi

Introduction xiii

1 Higher Order Derivatives 1

1.1 Divided Differences of Order n 1

1.2 General Derivatives of Order n 5

1.3 Generalized Riemann Derivatives of Order n 9

1.4 Peano Derivatives 16

1.4.1 Bilateral Peano Derivatives 16

1.4.2 Unilateral Peano Derivatives 19

1.4.3 Peano Boundedness 20

1.4.4 Generalized Peano Derivatives 21

1.4.5 Absolute Peano Derivatives 23

1.5 Riemann* Derivatives 24

1.5.1 Bilateral Riemann* Derivatives 24

1.5.2 Unilateral Riemann* Derivatives 27

1.6 Symmetric de la Vallée Poussin Derivatives 28

1.6.1 Symmetric de la Vallée Poussin Derivative and Symmetric Continuity 28

1.6.2 Smoothness 30

1.6.3 de la Vallée Poussin Boundedness 31

1.7 Symmetric Riemann* Derivatives 33

1.8 Cesàro Derivatives 33

1.8.1 Cesàro Continuity and Cesàro Derivative 34

1.8.2 Cesàro Boundedness 38

1.9 Symmetric Cesàro Derivatives 40

1.9.1 Symmetric Cesàro Continuity and Symmetric Cesàro Derivative 40

1.9.2 Symmetric Cesàro Boundedness 43

1.10 Borel Derivatives 44

1.10.1 Bilateral Borel Derivatives 44

1.10.2 Unilateral Borel Derivatives 47

1.10.3 Borel Boundedness 47

1.11 Symmetric Borel Derivatives 49

1.11.1 Symmetric Borel Derivatives and Symmetric Borel Continuity 49

1.11.2 Borel Smoothness 52

1.11.3 Symmetric Borel Boundedness 53

1.12 L^p-Derivatives 55

1.12.1 L^p-Derivatives and L^p-Continuity 55

1.12.2 L^p-Boundedness 59

1.13 Symmetric L^p-Derivatives 60

1.13.1 Symmetric L^p-Derivatives and Symmetric L^p-Continuity 60

1.13.2 L^p-Smoothness 62

1.13.3 Symmetric L^p-Boundedness 63

1.14 Abel Derivatives 63

1.14.1 Abel Summability 63

1.14.2 Abel Derivatives 64

1.14.3 Abel Continuity 68

1.14.4 Abel Smoothness 70

1.15 Laplace Derivatives 70

1.15.1 Laplace Derivatives and Laplace Continuity 70

1.15.2 Laplace Boundedness 75

1.15.3 Bilateral Laplace Derivatives 76

1.16 Symmetric Laplace Derivatives and Laplace Smoothness 77

1.16.1 Symmetric Laplace Derivatives and Symmetric Laplace Continuity 77

1.16.2 Symmetric Laplace Boundedness 82

1.16.3 Laplace Smoothness 84

2 Relations between Derivatives 85

2.1 Ordinary and Peano Derivatives, f^(k) and f^(k) 85

2.2 Riemann* and Peano Derivatives, f*_(k)and f_(k) 86

2.3 Symmetric Riemann* and Symmetric de la Vallée Poussin Derivatives, f*^(s)_(k) and f^(s)_(k) 88

2.4 Cesàro and Peano Derivatives 98

2.4.1 Cesàro and Peano derivatives, C_kDf and f_(k) 98

2.4.2 Cesàro and absolute Peano derivatives, C_kDf and f* 101

2.5 Peano and Symmetric de la Vallée Poussin Derivatives, f_(k) and f^(s)_(k), and Smoothness of Order k 102

2.6 Symmetric Cesàro and Symmetric de la Vallée Poussin Derivatives, SC_kDf and f^(s)_(k) 106

2.7 Borel and Peano Derivatives, BD_kf and f_(k) 108

2.8 Symmetric Borel and Symmetric de la Vallée Poussin Derivatives, SBD_kf and f^(s)_(k) 116

2.9 Borel and Symmetric Borel Derivatives, BD_kf and SBD_kf, and Borel Smoothness of Order k 127

2.10 Peano and L^p-Derivatives, f_(k) and f_(k),p 130

2.11 L^p- and Symmetric L^p-Derivatives, f_(k),p and f^(s)_(k),p 133

2.12 Symmetric de la Vallée Poussin and Symmetric L^p-Derivatives, f^(s)_(k) and f^(s)_(k),p 136

2.13 Borel and L^p-Derivatives, BD_(k)f and f_(k),p 140

2.14 Symmetric Borel and Symmetric L^p-Derivatives, SBD_kf and f^(s)_(k),p 144

2.15 Cesàro and Borel Derivatives, C_kDf and BD_kf 146

2.16 Symmetric Cesàro and Symmetric Borel Derivatives, SC_kDf and SBD_kf 149

2.17 Abel and Symmetric de la Vallée Poussin Derivatives, AD_kf and f^(s)_(k) 151

2.18 Laplace, Peano and Generalized Peano Derivatives, LD_kf, f_(k) and f_[k] 165

2.19 Laplace and Borel Derivatives, LD_kf and BD_kf 169

2.20 Symmetric Laplace and Symmetric de la Vallée Poussin Derivatives, SLD_kf and f^(s)_(k) 172

2.21 Laplace and Symmetric Laplace Derivatives, LD_kf and SLD_kf 175

2.22 Peano and Unsymmetric Riemann Derivatives, f_(k) and RD_kf 176

2.23 Symmetric de la Vallée Poussin and Symmetric Riemann Derivatives, f^(s)_(k) and RD^(s)_kf 178

2.24 Generalized Riemann and Peano Derivatives, GRD_kf and f_(k) 182

2.25 MZ- and Peano Derivatives, D_kf and f_(k) 184

Bibliography 187

Index 199