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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, Takagi–van 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 s ymbols, 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.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Functions, Continuous. --- Differentiable functions. --- Functions, Differentiable --- Continuous functions --- Mathematics. --- Fourier analysis. --- Functional analysis. --- Functions of real variables. --- Real Functions. --- Fourier Analysis. --- Functional Analysis. --- Functions of real variables --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Real variables --- Functions of complex variables

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This book gives a coherent account of the theory of Banach spaces and Banach lattices, using the spaces C_0(K) of continuous functions on a locally compact space K as the main example. The study of C_0(K) has been an important area of functional analysis for many years. It gives several new constructions, some involving Boolean rings, of this space as well as many results on the Stonean space of Boolean rings. The book also discusses when Banach spaces of continuous functions are dual spaces and when they are bidual spaces.

Mathematics. --- K-theory. --- Algebra. --- Ordered algebraic structures. --- Algebraic topology. --- Order, Lattices, Ordered Algebraic Structures. --- Algebraic Topology. --- K-Theory. --- Banach spaces. --- Functions, Continuous. --- Continuous functions --- Functions of complex variables --- Generalized spaces --- Topology --- Algebraic topology --- Homology theory --- Mathematics --- Mathematical analysis --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Algebra

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Digital Functions and Data Reconstruction: Digital-Discrete Methods provides a solid foundation to the theory of digital functions and its applications to image data analysis, digital object deformation, and data reconstruction. This new method has a unique feature in that it is mainly built on discrete mathematics with connections to classical methods in mathematics and computer sciences. Digitally continuous functions and gradually varied functions were developed in the late 1980s. A. Rosenfeld (1986) proposed digitally continuous functions for digital image analysis, especially to describe the “continuous” component in a digital image, which usually indicates an object. L. Chen (1989) invented gradually varied functions to interpolate a digital surface when the boundary appears to be continuous. In theory, digitally continuous functions are very similar to gradually varied functions. Gradually varied functions are more general in terms of being functions of real numbers; digitally continuous functions are easily extended to the mapping from one digital space to another.  This will be the first book about digital functions, which is an important modern research area for digital images and digitalized data processing, and provides an introduction and comprehensive coverage of digital function methods. Digital Functions and Data Reconstruction: Digital-Discrete Methods offers scientists and engineers who deal with digital data a highly accessible, practical, and mathematically sound introduction to the powerful theories of digital topology and functional analysis, while avoiding the more abstruse aspects of these topics.

Combinatorial analysis. --- Digital electronics. --- Machine theory. --- Number theory. --- Computer science --- Image processing --- Data recovery (Computer science) --- Engineering & Applied Sciences --- Electrical & Computer Engineering --- Mathematics --- Physical Sciences & Mathematics --- Applied Physics --- Technology - General --- Calculus --- Electrical Engineering --- Digital techniques --- Functions, Continuous. --- Data reconstruction (Computer science) --- Reconstruction, Data (Computer science) --- Recovery, Data (Computer science) --- Continuous functions --- Computer science. --- Computer graphics. --- Discrete mathematics. --- Computer Science. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Signal, Image and Speech Processing. --- Discrete Mathematics. --- Electronic data processing --- Computer vision. --- Machine vision --- Vision, Computer --- Artificial intelligence --- Pattern recognition systems --- Optical data processing. --- Signal processing. --- Image processing. --- Speech processing systems. --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Optical equipment