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Frames (Vector analysis)

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This book establishes brand-new frame theory and technical implementation in data science, with a special focus on spatial-scale feature extraction, network dynamics, object-oriented analysis, data-driven environmental prediction, and climate diagnosis. Given that data science is unanimously recognized as a core driver for achieving Sustainable Development Goals of the United Nations, these frame techniques bring fundamental changes to multi-channel data mining systems and support the development of digital Earth platforms. This book integrates the authors' frame research in the past twenty years and provides cutting-edge techniques and depth for scientists, professionals, and graduate students in data science, applied mathematics, environmental science, and geoscience. .

Artificial intelligence --- Mathematics. --- Bioclimatology. --- Environment. --- Data Science. --- Climate Change Ecology. --- Environmental Sciences. --- Data processing. --- Frames (Vector analysis) --- Ecology.

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This state-of-the-art textbook examines four research directions in harmonic analysis and features some of the latest applications in the field, including cosmic microwave background analysis, human cortex image denoising, and wireless communication. The work is the first one that combines spline theory (from a numerical or approximation-theoretical view), wavelets, frames, and time-frequency methods leading up to a construction of wavelets on manifolds other than Rn. Written by internationally renowned mathematicians, the interdisciplinary chapters are expository by design, enabling the reader to understand the theory behind modern image and signal processing methodologies. The main emphasis throughout the book is on the interdependence of the four modern research directions covered. Each chapter ends with exercises that allow for a more in-depth understanding of the material and are intended to stimulate the reader toward further research. A comprehensive index completes the work. Topics covered: * Frames and bases in mathematics and engineering * Wavelets with composite dilations and their applications * Wavelets on the sphere and their applications * Wiener's Lemma: theme and variations Four Short Courses on Harmonic Analysis is intended as a graduate-level textbook for courses or seminars on harmonic analysis and its applications. The work is also an excellent reference or self-study guide for researchers and practitioners with diverse mathematical backgrounds working in different fields such as pure and applied mathematics, image and signal processing engineering, mathematical physics, and communication theory.

Electronic books. -- local. --- Frames (Vector analysis). --- Harmonic analysis. --- Wavelets (Mathematics). --- Harmonic analysis --- Civil & Environmental Engineering --- Operations Research --- Engineering & Applied Sciences --- Wavelets (Mathematics) --- Frames (Vector analysis) --- Frame theory (Vector analysis) --- Wavelet analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematics. --- Fourier analysis. --- Physics. --- Fourier Analysis. --- Abstract Harmonic Analysis. --- Signal, Image and Speech Processing. --- Theoretical, Mathematical and Computational Physics. --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Vector analysis --- Analysis, Fourier --- Signal processing. --- Image processing. --- Speech processing systems. --- Mathematical physics. --- Physical mathematics --- Physics --- 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 --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication)

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The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature. Key features: * Basic results presented in an accessible way for both pure and applied mathematicians * Extensive exercises make the work suitable as a textbook for use in graduate courses * Full proofs included in introductory chapters; only basic knowledge of functional analysis required * Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series * Selected research topics presented with recommendations for more advanced topics and further reading * Open problems to simulate further research An Introduction to Frames and Riesz Basis will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.

Frames (Vector analysis) --- Bases (Linear topological spaces) --- Signal processing --- Mathematics. --- Bases (espaces vectoriels topologiques) --- Functional analysis. --- Applied mathematics. --- Engineering mathematics. --- Operator theory. --- Signal processing. --- Image processing. --- Speech processing systems. --- Functional Analysis. --- Applications of Mathematics. --- Operator Theory. --- Signal, Image and Speech Processing. --- 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 --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Functional analysis --- Engineering --- Engineering analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Signal processing - Mathematics. --- Acqui 2006 --- Analyse de fourier --- Ondelettes

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During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classical sufficient conditions for obtaining a frame. Consequently, there is a need for an updated book on frames, which moves the focus from the classical approach to a more constructive one. Based on a streamlined presentation of the author's previous work, An Introduction to Frames and Riesz Bases, this new textbook fills a gap in the literature, developing frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field. Key features and topics: * Results presented in an accessible way for graduate students, pure and applied mathematicians as well as engineers. * An introductory chapter provides basic results in finite-dimensional vector spaces, enabling readers with a basic knowledge of linear algebra to understand the idea behind frames without the technical complications in infinite-dimensional spaces. * Extensive exercises for use in theoretical graduate courses on bases and frames, or applications-oriented courses focusing on either Gabor analysis or wavelets. * Detailed description of frames with full proofs, an examination of the relationship between frames and Riesz bases, and a discussion of various ways to construct frames. * Content split naturally into two parts: The first part describes the theory on an abstract level, whereas the second part deals with explicit constructions of frames with applications and connections to time-frequency analysis, Gabor analysis, and wavelets. Frames and Bases: An Introductory Course will be an excellent textbook for graduate students as well as a good reference for researchers working in pure and applied mathematics, mathematical physics, and engineering. Practitioners working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find the book a useful self-study resource.

Stochastic processes --- Analytical topology --- Frames (Vector analysis) --- Bases (Linear topological spaces) --- Signal processing --- Cadres (Analyse vectorielle) --- Bases (Espaces vectoriels topologiques) --- Traitement du signal --- Mathematics. --- Mathématiques --- Mathematics --- Bases (Linear topological spaces). --- Frames (Vector analysis). --- Signal processing -- Mathematics. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Basis theory (Linear topological spaces) --- Frame theory (Vector analysis) --- Harmonic analysis. --- Fourier analysis. --- Functional analysis. --- Operator theory. --- Mathematical models. --- Functional Analysis. --- Fourier Analysis. --- Mathematical Modeling and Industrial Mathematics. --- Signal, Image and Speech Processing. --- Abstract Harmonic Analysis. --- Operator Theory. --- Linear topological spaces --- Sequences (Mathematics) --- Vector analysis --- Functional analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Analysis, Fourier --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 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 --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Models, Mathematical --- Simulation methods --- Signal processing - Mathematics