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

by Garg, Krishna M.

• ISBN13: 9780471253877
• ISBN10: 0471253871
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-09-23
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

Krishna M. Garg is the author of Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives, published by Wiley.

Table of Contents

General Introduction	1	(6)
PART I A UNIFIED THEORY OF UNILATERAL DERIVATES	7	(46)
Introduction to Part I	7	(2)
Chapter 1 Definitions, Notations, and Preliminary Results	9	(14)
1.1 Definitions and notations	9	(3)
1.2 Weaker forms of continuity and internal properties	12	(3)
1.3 Generalized and inherent properties of functions	15	(1)
1.4 Lower and upper Baire classes and relative measurability of functions	16	(3)
1.5 Baire classes and measurability of multifunctions	19	(4)
Chapter 2 Some Fundamental Theorems on Unilateral Derivates	23	(14)
2.1 Unilateral monotonicity and Lipschitz properties on a set	23	(2)
2.2 Two fundamental theorems on unilateral derivates	25	(4)
2.3 Extension of two theorems of G. C. Young and W. H. Young	29	(2)
2.4 Properties on some portion in terms of unilateral derivates	31	(1)
2.5 Extension of some theorems of W. H. Young and A. Denjoy	32	(5)
Chapter 3 Baire Class and Measurability of Derivates and Medians	37	(16)
3.1 A general theorem on unilateral derivates	37	(1)
3.2 Baire class and measurability of unilateral derivates	38	(5)
3.3 Baire class and measurability of medians	43	(2)
3.4 Relations between unilateral and strong derivates and medians	45	(3)
3.5 Refinement of a theorem of A. P. Morse	48	(5)
PART II A THEORY OF SOME NEW DERIVATIVES	53	(108)
Introduction to Part II	53	(4)
Chapter 4 Definitions of New Derivatives and Preliminary Results	57	(20)
4.1 Lower, upper, and semi-derivatives on normed vector spaces	57	(6)
4.2 Lower, upper, semi-, and weak derivatives on the real line	63	(3)
4.3 New derivatives and knotted and normal new derivatives; new derivability sets	66	(4)
4.4 Uniqueness of new derivatives	70	(4)
4.5 Continuity under new derivabilities	74	(3)
Chapter 5 Existence of New Derivatives	77	(12)
5.1 Lower derivability of LSC functions	77	(2)
5.2 Lower and upper strong bounded variation on a set	79	(3)
5.3 Lower differentiability of GL-BV(*) functions	82	(2)
5.4 Lower and upper strongly monotonic type and Lipschitz properties on a set	84	(3)
5.5 Normal lower derivability outside lower knot set; Denjoy-Young-Saks theorem	87	(2)
Chapter 6 Normal Level Structure and Other Derivability Theorems	89	(26)
6.1 Normal level structure theorem; two theorems of Saks	90	(3)
6.2 Normal lower derivability of continuous functions	93	(2)
6.3 Derivability of generalized regular and singular functions	95	(4)
6.4 Second level structure theorem	99	(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