Theory of Differentiation A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives

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ISBN: 9780471253877 | 0471253871
Cover: Hardcover
Copyright: 9/23/1998

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Theory of differentiation includes all aspects of various kinds of derivates and derivatives, and the theory of various Perron and Denjoy-Perron integrals. Derivative theorems covered are theorems on unilateral (or Dini) derivates. Through a cohesive format, outstanding problems are resolved, new ones are presented, and developments in this field, both past and present, are covered.

General Introduction

(6)

PART I A UNIFIED THEORY OF UNILATERAL DERIVATES

(46)

Introduction to Part I

(2)

Chapter 1 Definitions, Notations, and Preliminary Results

(14)

1.1 Definitions and notations

(3)

1.2 Weaker forms of continuity and internal properties

(3)

1.3 Generalized and inherent properties of functions

(1)

1.4 Lower and upper Baire classes and relative measurability of functions

(3)

1.5 Baire classes and measurability of multifunctions

(4)

Chapter 2 Some Fundamental Theorems on Unilateral Derivates

(14)

2.1 Unilateral monotonicity and Lipschitz properties on a set

(2)

2.2 Two fundamental theorems on unilateral derivates

(4)

2.3 Extension of two theorems of G. C. Young and W. H. Young

(2)

2.4 Properties on some portion in terms of unilateral derivates

(1)

2.5 Extension of some theorems of W. H. Young and A. Denjoy

(5)

Chapter 3 Baire Class and Measurability of Derivates and Medians

(16)

3.1 A general theorem on unilateral derivates

(1)

3.2 Baire class and measurability of unilateral derivates

(5)

3.3 Baire class and measurability of medians

(2)

3.4 Relations between unilateral and strong derivates and medians

(3)

3.5 Refinement of a theorem of A. P. Morse

(5)

PART II A THEORY OF SOME NEW DERIVATIVES

(108)

Introduction to Part II

(4)

Chapter 4 Definitions of New Derivatives and Preliminary Results

(20)

4.1 Lower, upper, and semi-derivatives on normed vector spaces

(6)

4.2 Lower, upper, semi-, and weak derivatives on the real line

(3)

4.3 New derivatives and knotted and normal new derivatives; new derivability sets

(4)

4.4 Uniqueness of new derivatives

(4)

4.5 Continuity under new derivabilities

(3)

Chapter 5 Existence of New Derivatives

(12)

5.1 Lower derivability of LSC functions

(2)

5.2 Lower and upper strong bounded variation on a set

(3)

5.3 Lower differentiability of GL-BV(*) functions

(2)

5.4 Lower and upper strongly monotonic type and Lipschitz properties on a set

(3)

5.5 Normal lower derivability outside lower knot set; Denjoy-Young-Saks theorem

(2)

Chapter 6 Normal Level Structure and Other Derivability Theorems

(26)

6.1 Normal level structure theorem; two theorems of Saks

(3)

6.2 Normal lower derivability of continuous functions

(2)

6.3 Derivability of generalized regular and singular functions

(4)

6.4 Second level structure theorem

(2)

6.5 Derivability in terms of properties of level sets

101

(4)

6.6 Ordinary derivability under properties (T(2)), (N), and (S); some theorems of Banach and Saks

105

(5)

6.7 Length of graph; Kolmogoroff-Vercenko theorem

110

(5)

Chapter 7 Calculus of New Derivatives

115

(46)

7.1 Similar, dissimilar, and compatible semi-and weak derivatives

116

(2)

7.2 New derivatives of restrictions of functions

118

(1)

7.3 Additivity and linearity of new derivatives

118

(7)

7.4 Product and quotient rules for new derivatives

125

(7)

7.5 Chain rules in terms of ordinary and new derivatives

132

(7)

7.6 General chain rules for new derivatives

139

(10)

7.7 Calculus of semiderivatives on normed vector spaces

149

(6)

7.8 Development in other fields related with new derivatives

155

(6)

PART III THEORY OF NEW DERIVATIVES (CONTINUED)

161

(114)

Introduction to Part III

161

(2)

Chapter 8 Mean Value Theorems and Related Results in Terms of New Derivatives

163

(26)

8.1 Mean value theorem in terms of new derivatives

163

(3)

8.2 Cauchy's mean value theorem and l'Hopital's rule in terms of new derivatives

166

(5)

8.3 Strong derivative in terms of limits of new derivatives

171

(2)

8.4 Taylor's formulae for new derivatives

173

(2)

8.5 Term-by-term differentiation for unilateral and strong derivatives

175

(5)

8.6 Term-by-term differentiation for new derivatives

180

(9)

Chapter 9 Monotonicity and Other Properties in Terms of New Derivatives

189

(26)

9.1 Monotonicity in terms of new derivatives

190

(5)

9.2 Points of extrema and concavity in terms of new derivatives

195

(3)

9.3 Monotonicity under properties (T(2)) and (N) in terms of ordinary derivative

198

(3)

9.4 Medians at nonmonotonicity points

201

(5)

9.5 Properties of lower and weakly derivable functions

206

(1)

9.6 Monotonicity in terms of medians and Goldowski-Tonelli theorems in terms of new derivatives

207

(8)

Chapter 10 Properties of New Derivatives and New Derivability Sets

215

(20)

10.1 Borel class and measurability of new derivability and knot sets

215

(6)

10.2 Zahorski property of new derivability sets

221

(4)

10.3 Baire class and measurability of new derivatives and medians

225

(3)

10.4 Darboux and Denjoy properties of new derivatives

228

(5)

10.5 Stationary sets of new derivatives

233

(2)

Chapter 11 Denjoy and Perron Integrals Corresponding to New Derivatives

235

(40)

11.1 Perron integrals corresponding to new derivatives

235

(4)

11.2 Elementary properties of new Perron integrals

239

(4)

11.3 Properties of new indefinite Perron integrals

243

(5)

11.4 Lower and upper AC, L-AC(*), U-AC(*), and AC(*) functions

248

(7)

11.5 Criteria for GL-AC(*) and GL-BV(*) in terms of derivates

255

(3)

11.6 Descriptive definitions of lower and upper Perron integrals

258

(11)

11.7 Semistrong properties S-BV(*) and S-AC(*)

269

(2)

11.8 Descriptive definition of weak Perron integral

271

(4)

PART IV SOME DIRECT APPLICATIONS OF THE THEORY OF NEW DERIVATIVES

275

(54)

Introduction to Part IV

275

(2)

Chapter 12 Derivates and Derivability of Symmetric, Quasismooth and Smooth Functions

277

(20)

12.1 Lower and upper symmetric, quasismooth and smooth functions

278

(1)

12.2 Continuity of symmetric and quasismooth functions

279

(4)

12.3 Symmetry of derivates of symmetric and lower smooth functions

283

(3)

12.4 Ordinary derivability of quasismooth functions

286

(4)

12.5 Properties of quasismooth functions in terms of derivates and ordinary derivative

290

(3)

12.6 Differentiability of smooth functions and their properties in terms of derivates and ordinary derivative

293

(3)

12.7 Properties of ordinary derivative of smooth functions

296

(1)

Chapter 13 Differential Structure at Nonmonotonicity Points and Nowhere Monotone and Nonderivable Functions

297

(18)

13.1 Differential structure at nonmonotonicity points

298

(3)

13.2 Nowhere monotone functions

301

(1)

13.3 Strongly nowhere monotone functions

302

(2)

13.4 Differential structure of lower singular functions at nonmonotonicity points

304

(3)

13.5 The nature of new derivatives of nonderivable functions

307

(5)

13.6 New derivatives of Brownian paths

312

(3)

Chapter 14 Strong Derivates and Strong Derivative

315

(14)

14.1 Properties of strong derivates, strong median, and strong derivative

316

(2)

14.2 Local and global properties of functions in terms of strong derivates

318

(4)

14.3 Relations between unilateral and strong derivates; Dini's theorems

322

(1)

14.4 Characterizations of strong derivability

323

(3)

14.5 Mean value property and the Darboux and Denjoy properties of strong derivative

326

(3)

PART V UNIFIED AXIOMATIC THEORIES OF GENERALIZED DERIVATIVES

329

(114)

Introduction to Part V

329

(4)

Chapter 15 An Axiomatic Model of Generalized Limits and Derivates

333

(24)

15.1 Admissible and s- and s(*) -admissible generalized limits

333

(5)

15.2 Admissible and s- and s(*) -admissible generalized quotients and derivates

338

(5)

15.3 Examples of admissible generalized limits and derivates

343

(8)

15.4 Regular, subregular, and totally regular generalized derivates

351

(4)

15.5 s(1)-admissible generalized limits

355

(2)

Chapter 16 A Unified Theory of Generalized Derivatives

357

(28)

16.1 Comparison of new derivatives with other generalized derivatives

358

(4)

16.2 Compatibility of admissible generalized derivatives with new derivatives

362

(1)

16.3 Calculus of generalized derivatives

363

(8)

16.4 Mean value theorems and l'Hopital's rule in terms of generalized derivatives

371

(1)

16.5 Monotonicity in terms of generalized derivates and derivatives; Goldowski-Tonelli theorem

372

(4)

16.6 Relations between generalized and strong derivates

376

(1)

16.7 Properties of generalized derivatives

377

(4)

16.8 New generalized derivates and derivatives

381

(4)

Chapter 17 A Unified Theory of Generalized Symmetric Derivatives

385

(34)

17.1 Admissible and regular generalized symmetric derivates

386

(10)

17.2 Conditional compatibility of generalized symmetric derivatives with new derivatives

396

(3)

17.3 Calculus of generalized symmetric derivatives

399

(5)

17.4 Mean value theorems and l'Hopital's rule in terms of generalized symmetric derivatives

404

(3)

17.5 Monotonicity and Goldowski-Tonelli theorems in terms of generalized symmetric derivatives

407

(3)

17.6 Relations between generalized symmetric derivates and strong derivates

410

(1)

17.7 Properties of generalized symmetric derivatives

411

(3)

17.8 New generalized symmetric derivates and derivatives

414

(5)

Chapter 18 A Unified Theory of Generalized New Derivatives

419

(24)

18.1 Generalized lower, upper, semi-, and weak derivatives

419

(3)

18.2 Derivability theorems for generalized new derivatives

422

(1)

18.3 Calculus of generalized new derivatives

423

(9)

18.4 Mean value theorems and l'Hopital's rule in terms of generalized new derivatives

432

(3)

18.5 Monotonicity and Goldowski-Tonelli theorems in terms of generalized new derivatives

435

(2)

18.6 Relations between generalized new derivatives and strong derivative

437

(1)

18.7 Properties of generalized new derivatives

437

(6)

PART VI UNIFIED THEORIES OF SOME OTHER ASPECTS OF GENERALIZED DERIVATES AND DERIVATIVES

443

(60)

Introduction to Part VI

443

(2)

Chapter 19 A Unified Theory of Generalized Derivates of Typical Continuous Functions

445

(24)

19.1 Connected generalized derivates on C

445

(7)

19.2 Generalized derivates of typical continuous functions

452

(4)

19.3 Generalized symmetric derivates of typical continuous functions

456

(3)

19.4 Generalized nonderivability and generalized symmetric nonderivability of typical continuous functions

459

(4)

19.5 Generalized knotted semiderivability and unilateral derivability of typical continuous functions

463

(6)

Chapter 20 A Unified Theory of Generalized Smooth Functions

469

(8)

20.1 Generalized smooth functions and admissibility of generalized smoothness

469

(3)

20.2 Generalized derivability of generalized smooth functions

472

(2)

20.3 Mean value and Darboux properties of generalized derivatives of generalized smooth functions

474

(3)

Chapter 21 A Unified Axiomatic Theory of Generalized Perron Integrals

477

(26)

21.1 H-admissible generalized derivates and generalized Perron integrals

477

(5)

21.2 P-admissible and P-regular generalized Perron integrals

482

(7)

21.3 Elementary properties of generalized Perron integrals

489

(4)

21.4 Properties of indefinite generalized Perron integrals

493

(5)

21.5 Applications of the axiomatic theory and open problems

498

(5)

References

503

(12)

Index of Symbols

515

(6)

Subject Index

521

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Theory of Differentiation A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives

Summary

Author Biography

Table of Contents

Supplemental Materials

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Theory of Differentiation A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives

Summary

Author Biography

Table of Contents

Supplemental Materials

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