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 Riemann's continuous, and purportedly nowhere differentiable, function. For the reader's 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 book's 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.

Provides a systematic reference for research into, and courses on, the subject of continuous nowhere differentiable functions Anthologizes over 150 years of progress on the topic of continuous nowhere differentiable functions Consolidates an array of established results into a single, self-contained volume including detailed proofs Includes supplementary material: sn.pub/extras

Autorentext

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.

Leseprobe
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Inhalt

Preface.- Preliminaries.- Weierstrass type functions I.- Takagivan der Waerden type functions I.- Bolzano type functions I.- Other examples.- Baire category approach.- Weierstrass type functions II.- Takagivan der Waerden type functions II.- Bolzano type functions II.- BesicovitchMorse Functions.- Linear spaces of nowhere differentiable functions.- Riemann function.