Theory of differentiation: a unified theory of differentiation via new derivate theorems and new derivatives
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1998
|
| Series: | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts
24 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 525 S. |
| ISBN: | 0471253871 |
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MARC
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| 100 | 1 | |a Garg, Krishna M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Theory of differentiation |b a unified theory of differentiation via new derivate theorems and new derivatives |c Krishna M. Garg |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1998 | |
| 300 | |a XV, 525 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |v 24 | |
| 490 | 0 | |a A Wiley-Interscience publication | |
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| 650 | 7 | |a Calcul différentiel |2 ram | |
| 650 | 4 | |a Differential calculus | |
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Record in the Search Index
| _version_ | 1804126931641171968 |
|---|---|
| adam_text | Contents
General Introduction 1
PART I A UNIFIED THEORY OF UNILATERAL DERIVATES 7
Introduction to Part I 7
Chapter 1 Definitions, Notations, and Preliminary Results 9
§1.1 Definitions and notations 9
§1.2 Weaker forms of continuity and internal properties 12
§1.3 Generalized and inherent properties of functions 15
§1.4 Lower and upper Baire classes and relative measurability
of functions 16
§1.5 Baire classes and measurability of multifunctions 19
Chapter 2 Some Fundamental Theorems on Unilateral Derivates 23
§2.1 Unilateral monotonicity and Lipschitz properties on a set 23
§2.2 Two fundamental theorems on unilateral derivates 25
§2.3 Extension of two theorems of G. C. Young and W. H.
Young 29
§2.4 Properties on some portion in terms of unilateral
derivates 31
§2.5 Extension of some theorems of W. H. Young and
A. Denjoy 32
Chapter 3 Baire Class and Measurability of Derivates and Medians 37
§3.1 A general theorem on unilateral derivates 37
§3.2 Baire class and measurability of unilateral derivates 38
§3.3 Baire class and measurability of medians 43
ix
x Contents
§3.4 Relations between unilateral and strong derivates and
medians 45
§3.5 Refinement of a theorem of A. P. Morse 48
PART II A THEORY OF SOME NEW DERIVATIVES 53
Introduction to Part II 53
Chapter 4 Definitions of New Derivatives and Preliminary Results 57
§4.1 Lower, upper, and semi derivatives on normed vector
spaces 57
§4.2 Lower, upper, semi , and weak derivatives on the real
line 63
§4.3 New derivatives and knotted and normal new derivatives;
new derivability sets 66
§4.4 Uniqueness of new derivatives 70
§4.5 Continuity under new derivabilities 74
Chapter 5 Existence of New Derivatives 77
§5.1 Lower derivability of LSC functions 77
§5.2 Lower and upper strong bounded variation on a set 79
§5.3 Lower differentiability of GL BV* functions 82
§5.4 Lower and upper strongly monotonic type and Lipschitz
properties on a set 84
§5.5 Normal lower derivability outside lower knot set;
Denjoy Young Saks theorem 87
Chapter 6 Normal Level Structure and Other Derivability Theorems 89
§6.1 Normal level structure theorem; two theorems of Saks 90
§6.2 Normal lower derivability of continuous functions 93
§6.3 Derivability of generalized regular and singular functions 95
§6.4 Second level structure theorem 99
§6.5 Derivability in terms of properties of level sets 101
§6.6 Ordinary derivability under properties (T2), (N), and
(5); some theorems of Banach and Saks 105
§6.7 Length of graph; Kolmogoroff Vercenko theorem 110
Chapter 7 Calculus of New Derivatives 115
§7.1 Similar, dissimilar, and compatible semi and weak
derivatives 116
§7.2 New derivatives of restrictions of functions 118
§7.3 Additivity and linearity of new derivatives 118
§7.4 Product and quotient rules for new derivatives 125
Contents xi
§7.5 Chain rules in terms of ordinary and new derivatives 132
§7.6 General chain rules for new derivatives 139
§7.7 Calculus of semiderivatives on normed vector spaces 149
§7.8 Developments in other fields related with new derivatives 155
PART III THEORY OF NEW DERIVATIVES (CONTINUED) 161
Introduction to Part III 161
Chapter 8 Mean Value Theorems and Related Results in Terms of New
Derivatives 163
§8.1 Mean value theorem in terms of new derivatives 163
§8.2 Cauchy s mean value theorem and l Hopital s rule in
terms of new derivatives 166
§8.3 Strong derivative in terms of limits of new derivatives 171
§8.4 Taylor s formulae for new derivatives 173
§8.5 Term by term differentiation for unilateral and strong
derivatives 175
§8.6 Term by term differentiation for new derivatives 180
Chapter 9 Monotonicity and Other Properties in Terms of New
Derivatives 189
§9.1 Monotonicity in terms of new derivatives 190
§9.2 Points of extrema and concavity in terms of new
derivatives 195
§9.3 Monotonicity under properties (J2) and (N) in terms of
ordinary derivative 198
§9.4 Medians at nonmonotonicity points 201
§9.5 Properties of lower and weakly derivable functions 206
§9.6 Monotonicity in terms of medians and Goldowski Tonelli
theorems in terms of new derivatives 207
Chapter 10 Properties of New Derivatives and New Deducibility Sets 215
§10.1 Borel class and measurability of new derivability and
knot sets 215
§10.2 Zahorski property of new derivability sets 221
§10.3 Baire class and measurability of new derivatives and
medians 225
§10.4 Darboux and Denjoy properties of new derivatives 228
§10.5 Stationary sets of new derivatives 233
xii Contents
Chapter 11 Denjoy and Perron Integrals Corresponding to New
Derivatives 235
§11.1 Perron integrals corresponding to new derivatives 235
§11.2 Elementary properties of new Perron integrals 239
§11.3 Properties of new indefinite Perron integrals 243
§11.4 Lower and upper AC, L AC*, U AC*, and AC* functions 248
§11.5 Criteria for GL AQ and GL BV* in terms of derivates 255
§11.6 Descriptive definitions of lower and upper Perron
integrals 258
§11.7 Semistrong properties S BV* and S AC* 269
§11.8 Descriptive definition of weak Perron integral 271
PART IV SOME DIRECT APPLICATIONS OF THE THEORY OF
NEW DERIVATIVES 275
Introduction to Part IV 275
Chapter 12 Derivates and Derivability of Symmetric, Quasismooth and
Smooth Functions 277
§12.1 Lower and upper symmetric, quasismooth and smooth
functions 278
§12.2 Continuity of symmetric and quasismooth functions 279
§12.3 Symmetry of derivates of symmetric and lower smooth
functions 283
§12.4 Ordinary derivability of quasismooth functions 286
§12.5 Properties of quasismooth functions in terms of derivates
and ordinary derivative 290
§12.6 Differentiability of smooth functions and their properties
in terms of derivates and ordinary derivative 293
§12.7 Properties of ordinary derivative of smooth functions 296
Chapter 13 Differential Structure at Nonmonotonicity Points and
Nowhere Monotone and Nonderivable Functions 297
§13.1 Differential structure at nonmonotonicity points 298
§13.2 Nowhere monotone functions 301
§13.3 Strongly nowhere monotone functions 302
§13.4 Differential structure of lower singular functions at
nonmonotonicity points 304
§13.5 The nature of new derivatives of nonderivable functions 307
§13.6 New derivatives of Brownian paths 312
Chapter 14 Strong Derivates and Strong Derivative 315
§14.1 Properties of strong derivates, strong median, and strong
derivative 316
Contents xiii
§14.2 Local and global properties of functions in terms of
strong derivates 318
§14.3 Relations between unilateral and strong derivates; Dini s
theorems 322
§14.4 Characterizations of strong derivability 323
§14.5 Mean value property and the Darboux and Denjoy
properties of strong derivative 326
PART V UNIFIED AXIOMATIC THEORIES OF GENERALIZED
DERIVATIVES 329
Introduction to Part V 329
Chapter 15 An Axiomatic Model of Generalized Limits and Derivates 333
§15.1 Admissible and s and ^ admissible generalized limits 333
§15.2 Admissible and s and s* admissible generalized
quotients and derivates 338
§15.3 Examples of admissible generalized limits and derivates 343
§15.4 Regular, subregular, and totally regular generalized
derivates 351
§15.5 .?, admissible generalized limits 355
Chapter 16 A Unified Theory of Generalized Derivatives 357
§16.1 Comparison of new derivatives with other generalized
derivatives 358
§16.2 Compatibility of admissible generalized derivatives with
new derivatives 362
§16.3 Calculus of generalized derivatives 363
§16.4 Mean value theorems and l Hbpital s rule in terms of
generalized derivatives 371
§16.5 Monotonicity in terms of generalized derivates and
derivatives; Goldowski Tonelli theorem 372
§16.6 Relations between generalized and strong derivates 376
§16.7 Properties of generalized derivatives 377
§16.8 New generalized derivates and derivatives 381
Chapter 17 A Unified Theory of Generalized Symmetric Derivatives 385
§17.1 Admissible and regular generalized symmetric derivates 386
§17.2 Conditional compatibility of generalized symmetric
derivatives with new derivatives 396
§17.3 Calculus of generalized symmetric derivatives 399
§17.4 Mean value theorems and l Hopital s rule in terms of
generalized symmetric derivatives 404
xiv Contents
§17.5 Monotonicity and Goldowski Tonelli theorems in terms
of generalized symmetric derivatives 407
§17.6 Relations between generalized symmetric derivates and
strong derivates 410
§17.7 Properties of generalized symmetric derivatives 411
§17.8 New generalized symmetric derivates and derivatives 414
Chapter 18 A Unified Theory of Generalized New Derivatives 419
§18.1 Generalized lower, upper, semi , and weak derivatives 419
§18.2 Derivability theorems for generalized new derivatives 422
§18.3 Calculus of generalized new derivatives 423
§18.4 Mean value theorems and l Hopital s rule in terms of
generalized new derivatives 432
§18.5 Monotonicity and Goldowski Tonelli theorems in terms
of generalized new derivatives 435
§18.6 Relations between generalized new derivatives and strong
derivative 437
§18.7 Properties of generalized new derivatives 437
PART VI UNIFIED THEORIES OF SOME OTHER ASPECTS OF
GENERALIZED DERIVATES AND DERIVATIVES 443
Introduction to Part VI 443
Chapter 19 A Unified Theory of Generalized Derivates of Typical
Continuous Functions 445
§19.1 Connected generalized derivates on C 445
§19.2 Generalized derivates of typical continuous functions 452
§19.3 Generalized symmetric derivates of typical continuous
functions 456
§19.4 Generalized nonderivability and generalized symmetric
nonderivability of typical continuous functions 459
§19.5 Generalized knotted semiderivability and unilateral
derivability of typical continuous functions 463
Chapter 20 A Unified Theory of Generalized Smooth Functions 469
§20.1 Generalized smooth functions and admissibility of
generalized smoothness 469
§20.2 Generalized derivability of generalized smooth
functions 472
§20.3 Mean value and Darboux properties of generalized
derivatives of generalized smooth functions 474
Contents xv
Chapter 21 A Unified Axiomatic Theory of Generalized Perron Integrals All
§21.1 // admissible generalized derivates and generalized
Perron integrals 477
§21.2 / admissible and P regular generalized Perron integrals 482
§21.3 Elementary properties of generalized Perron integrals 489
§21.4 Properties of indefinite generalized Perron integrals 493
§21.5 Applications of the axiomatic theory and open problems 498
References 503
Index of Symbols 515
Subject Index 521
|
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| institution | BVB |
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| language | English |
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| physical | XV, 525 S. |
| publishDate | 1998 |
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| publisher | Wiley |
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| series | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |
| series2 | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts A Wiley-Interscience publication |
| spelling | Garg, Krishna M. Verfasser aut Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives Krishna M. Garg New York [u.a.] Wiley 1998 XV, 525 S. txt rdacontent n rdamedia nc rdacarrier Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts 24 A Wiley-Interscience publication Calcul différentiel Calcul différentiel ram Differential calculus Differentialrechnung (DE-588)4012252-9 gnd rswk-swf Differentialrechnung (DE-588)4012252-9 s DE-604 Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts 24 (DE-604)BV012067942 24 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008343422&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Garg, Krishna M. Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts Calcul différentiel Calcul différentiel ram Differential calculus Differentialrechnung (DE-588)4012252-9 gnd |
| subject_GND | (DE-588)4012252-9 |
| title | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives |
| title_auth | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives |
| title_exact_search | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives |
| title_full | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives Krishna M. Garg |
| title_fullStr | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives Krishna M. Garg |
| title_full_unstemmed | Theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives Krishna M. Garg |
| title_short | Theory of differentiation |
| title_sort | theory of differentiation a unified theory of differentiation via new derivate theorems and new derivatives |
| title_sub | a unified theory of differentiation via new derivate theorems and new derivatives |
| topic | Calcul différentiel Calcul différentiel ram Differential calculus Differentialrechnung (DE-588)4012252-9 gnd |
| topic_facet | Calcul différentiel Differential calculus Differentialrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008343422&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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