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Continuous Nowhere Differentiable Functions: The Monsters of Analysis 1st ed. 2015 [Kõva köide]

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  • Formaat: Hardback, 299 pages, kõrgus x laius: 254x178 mm, kaal: 822 g, 14 Illustrations, color; 1 Illustrations, black and white; XII, 299 p. 15 illus., 14 illus. in color., 1 Hardback
  • Sari: Springer Monographs in Mathematics
  • Ilmumisaeg: 13-Jan-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319126695
  • ISBN-13: 9783319126692
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Continuous Nowhere Differentiable Functions: The Monsters of Analysis 1st ed. 2015Suurem pilt
  • Formaat: Hardback, 299 pages, kõrgus x laius: 254x178 mm, kaal: 822 g, 14 Illustrations, color; 1 Illustrations, black and white; XII, 299 p. 15 illus., 14 illus. in color., 1 Hardback
  • Sari: Springer Monographs in Mathematics
  • Ilmumisaeg: 13-Jan-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319126695
  • ISBN-13: 9783319126692
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319126692.html
Märksõnad:
This book covers the construction, analysis, and theory of continuous nowhere differentiable functions, comprehensively and accessibly. After illuminating the significance of the subject through an overview of its history, the reader is introduced to the sophisticated toolkit of ideas and tricks used to study the explicit continuous nowhere differentiable functions of Weierstrass, Takagivan der Waerden, Bolzano, and others. Modern tools of functional analysis, measure theory, and Fourier analysis are applied to examine the generic nature of continuous nowhere differentiable functions, as well as linear structures within the (nonlinear) space of continuous nowhere differentiable functions. To round out the presentation, advanced techniques from several areas of mathematics are brought together to give a state-of-the-art analysis of Riemanns continuous, and purportedly nowhere differentiable, function.

For the readers benefit, claims requiring elaboration, and open problems, are clearly indicated. An appendix conveniently provides background material from analysis and number theory, and comprehensive indices of symbols, problems, and figures enhance the books utility as a reference work. Students and researchers of analysis will value this unique book as a self-contained guide to the subject and its methods.

Arvustused

This book is a thorough survey of a function that, at its announcement, took the mathematical world by storm. This book can be recommended for those whose research involves working in analysis. Of interest to many will be the open problems placed throughout the text, along with the extensive bibliography. a very detailed book which, if you work in classical analysis or topology or expect to work with students interested in this topic, belongs in your collection. (Robert W. Vallin, Mathematical Reviews, June, 2016)

By bringing together results scattered in various publications, some of them hardly to find or/and hardly to read (I mean old papers), presenting them in a unitary and rigorous way (using a modern language and style) with pertinent historical comments, the authors have done a great service to the mathematical community. The book presents interest for all mathematicians, but also for people (engineers, physicists, etc) having a basic background in calculus, interested in the evolution . (S. Cobza, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (1), 2016)

The book presents the construction, analysis, and theory of continuous nowhere differentiable functions in a comprehensive and accessible manner. This unique book will be of interest to students and researchers of analysis as a self-contained guide to the subject of continuous nowhere differentiable functions and its methods. (Zoltán Finta, zbMATH 1334.26001, 2016)

1 Introduction: A Historical Journey
1(6)
Part I Classical Results
7(98)
2 Preliminaries
9(10)
2.1 Derivatives
9(3)
2.2 Families of Continuous Nowhere Differentiable Functions
12(1)
2.3 The Denjoy--Young--Saks Theorem
12(3)
2.4 Series of Continuous Functions
15(1)
2.5 Holder Continuity
16(3)
3 Weierstrass-Type Functions I
19(32)
3.1 Introduction
19(3)
3.2 General Properties of Wp,a,b,θ
22(2)
3.3 Differentiability of Wp,a,b,θ (in the Infinite Sense)
24(2)
3.4 An Open Problem
26(1)
3.5 Weierstrass's Method
27(8)
3.5.1 Lerch's Results
31(2)
3.5.2 Porter's Results
33(2)
3.6 Cellerier's Method
35(1)
3.7 Dini's Method
36(2)
3.8 Bromwich's Method
38(1)
3.9 Behrend's Method
39(7)
3.10 Emde Boas's Method
46(2)
3.11 The Method of Baouche--Dubuc
48(1)
3.12 Summary
49(2)
4 Takagi--van der Waerden-Type Functions I
51(14)
4.1 Introduction
51(5)
4.2 Kairies's Method
56(2)
4.3 Cater's Method
58(3)
4.4 Differentiability of a Class of Takagi Functions
61(4)
5 Bolzano-Type Functions I
65(34)
5.1 The Bolzano-Type Function
65(4)
5.2 Q-Representation of Numbers
69(4)
5.2.1 Continuity of Functions Given via Q-Representation
70(1)
5.2.2 Bolzano-Type Functions Defined via Q-Representation
71(2)
5.3 Examples of Bolzano-Type Functions
73(11)
5.3.1 The Hahn Function
73(1)
5.3.2 The Kiesswetter Function
73(6)
5.3.3 The Okamoto Function
79(5)
5.4 Continuity of Functions Given by Arithmetic Formulas
84(1)
5.5 Sierpinski Function
85(1)
5.6 The Pratsiovytyi--Vasylenko Functions
86(1)
5.7 Petr Function
87(2)
5.8 Wunderlich--Bush--Wen Function
89(2)
5.9 Wen Function
91(2)
5.10 Singh Functions
93(6)
6 Other Examples
99(6)
6.1 Schoenberg Functions
99(3)
6.2 Second Wen Function
102(3)
Part II Topological Methods
105(26)
7 Baire Category Approach
107(24)
7.1 Metric Spaces and First Baire Category
107(1)
7.2 The Banach--Jarnik--Mazurkiewicz Theorem
108(5)
7.3 Typical Functions in the Disk Algebra
113(2)
7.4 The Jarnik--Marcinkiewicz Theorems
115(6)
7.5 The Saks Theorem
121(3)
7.6 The Banach--Mazurkiewicz Theorem Revisited
124(3)
7.7 The Structure of ND(I)
127(4)
Part III Modern Approach
131(124)
8 Weierstrass-Type Functions II
133(54)
8.1 Introduction
133(1)
8.2 Hardy's Method
134(9)
8.3 Baouche--Dubuc Method
143(2)
8.4 Kairies--Girgensohn Method
145(12)
8.4.1 A System of Functional Equations
145(1)
8.4.2 The Faber--Schauder Basis of C(I)
146(3)
8.4.3 Nowhere Differentiability and the Schauder Coefficients
149(2)
8.4.4 Schauder Coefficients of Solutions of a System of Functional Equations
151(3)
8.4.5 Nowhere Differentiability of W1,a,b,θ for ab ≥ 1, b N2
154(3)
8.5 Weierstrass-Type Functions from a General Point of View
157(4)
8.6 Johnsen's Method
161(10)
8.7 Hata's Method
171(15)
8.7.1 Nowhere Differentiability of the Weierstrass-Type Functions: Finite One-Sided Derivatives
171(3)
8.7.2 Knot Points of Weierstrass-Type Functions
174(4)
8.7.3 Nowhere Differentiability of Weierstrass-Type Functions: Infinite Derivatives
178(8)
8.8 Summary
186(1)
9 Takagi--van der Waerden-Type Functions II
187(16)
9.1 Introduction
187(1)
9.2 The Case ab > 1
187(4)
9.3 Infinite Unilateral Derivatives of T1/2,2,0
191(7)
9.4 Proof of Theorem 9.3.4
198(2)
9.5 The Case of Normal Numbers
200(3)
10 Bolzano-Type Functions II
203(6)
10.1 Bolzano-Type Functions
203(6)
11 Besicovitch Functions
209(36)
11.1 Morse's Besicovitch Function
209(11)
11.1.1 Preparation
209(3)
11.1.2 A Class of Continuous Functions and Its Properties
212(1)
11.1.3 A New Function ƒ for Every ƒ
213(4)
11.1.4 A Besicovitch--Morse Function
217(3)
11.2 Singh's Besicovitch Function
220(15)
11.2.1 A Representation of Numbers
220(4)
11.2.2 Definition of Singh's Besicovitch Function
224(2)
11.2.3 Continuity of S4
226(2)
11.2.4 Nowhere Differentiability of S4
228(7)
11.3 BM(I) Is Residual in a Certain Subspace of C(I)
235(10)
12 Linear Spaces of Nowhere Differentiable Functions
245(10)
12.1 Introduction
245(1)
12.2 c-Lineability of ND∞(R)
246(2)
12.3 Spaceability of ND±(I)
248(7)
12.3.1 Two Matrices
248(1)
12.3.2 Auxiliary Functions
249(1)
12.3.3 The Closed Linear Subspace E ⊂ ND±(I)
250(5)
Part IV Riemann Function
255(10)
13 Riemann Function
257(8)
13.1 Introduction
257(1)
13.2 Auxiliary Lemmas
258(3)
13.3 Differentiability of the Riemann Function
261(4)
Appendix A
265(14)
A.1 Cantor Representation
265(1)
A.2 Harmonic and Holomorphic Functions
266(2)
A.3 Fourier Transform
268(2)
A.4 Fresnel Function
270(1)
A.5 Poisson Summation Formula
270(1)
A.6 Legendre, Jacobi, and Kronecker Symbols
271(2)
A.7 Gaussian Sums
273(1)
A.8 Farey Fractions
274(1)
A.9 Normal Numbers
275(4)
Appendix B List of Symbols
279(4)
B.1 General Symbols
279(1)
B.2 Symbols in Individual
Chapters
280(3)
Appendix C List of Problems
283(2)
List of Figures 285(2)
References 287(10)
Index 297
Marek Jarnicki is Professor of Mathematics at Jagiellonian University, Poland. His primary subject of research is complex analysis, particularly holomorphically invariant (contractible) pseudodistances and pseudometrics; domains of holomorphy with respect to special cases of holomorphic functions; continuation of holomorphic functions with restricted growth; and the extension of separately analytic functions.

Peter Pflug is Professor of Mathematics at the University of Oldenburg, Germany. His primary subject of research is the theory of functions of several complex variables and complex analysis.