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Autor Christensen, Ole |
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Título : Approximation Theory : From Taylor Polynomials to Wavelets Tipo de documento: documento electrónico Autores: Christensen, Ole ; SpringerLink (Online service) ; Christensen, Khadija L Editorial: Boston, MA : Birkhäuser Boston Fecha de publicación: 2005 Otro editor: Imprint: Birkhäuser Colección: Applied and Numerical Harmonic Analysis, ISSN 2296-5009 Número de páginas: XI, 156 p. 5 illus Il.: online resource ISBN/ISSN/DL: 978-0-8176-4448-2 Idioma : Inglés (eng) Palabras clave: Mathematics Harmonic analysis Approximation theory Fourier Functional Applied mathematics Engineering Analysis Approximations and Expansions Abstract Applications of Signal, Image Speech Processing Clasificación: 51 Matemáticas Resumen: This concisely written book gives an elementary introduction to a classical area of mathematics—approximation theory—in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Key features and topics: * Description of wavelets in words rather than mathematical symbols * Elementary introduction to approximation using polynomials (Weierstrass’ and Taylor’s theorems) * Introduction to infinite series, with emphasis on approximation-theoretic aspects * Introduction to Fourier analysis * Numerous classical, illustrative examples and constructions * Discussion of the role of wavelets in digital signal processing and data compression, such as the FBI’s use of wavelets to store fingerprints * Minimal prerequisites: elementary calculus * Exercises that may be used in undergraduate and graduate courses on infinite series and Fourier series Approximation Theory: From Taylor Polynomials to Wavelets will be an excellent textbook or self-study reference for students and instructors in pure and applied mathematics, mathematical physics, and engineering. Readers will find motivation and background material pointing toward advanced literature and research topics in pure and applied harmonic analysis and related areas Nota de contenido: 1 Approximation with Polynomials -- 1.1 Approximation of a function on an interval -- 1.2 Weierstrass’ theorem -- 1.3 Taylor’s theorem -- 1.4 Exercises -- 2 Infinite Series -- 2.1 Infinite series of numbers -- 2.2 Estimating the sum of an infinite series -- 2.3 Geometric series -- 2.4 Power series -- 2.5 General infinite sums of functions -- 2.6 Uniform convergence -- 2.7 Signal transmission -- 2.8 Exercises -- 3 Fourier Analysis -- 3.1 Fourier series -- 3.2 Fourier’s theorem and approximation -- 3.3 Fourier series and signal analysis -- 3.4 Fourier series and Hilbert spaces -- 3.5 Fourier series in complex form -- 3.6 Parseval’s theorem -- 3.7 Regularity and decay of the Fourier coefficients -- 3.8 Best N-term approximation -- 3.9 The Fourier transform -- 3.10 Exercises -- 4 Wavelets and Applications -- 4.1 About wavelet systems -- 4.2 Wavelets and signal processing -- 4.3 Wavelets and fingerprints -- 4.4 Wavelet packets -- 4.5 Alternatives to wavelets: Gabor systems -- 4.6 Exercises -- 5 Wavelets and their Mathematical Properties -- 5.1 Wavelets and L2 (?) -- 5.2 Multiresolution analysis -- 5.3 The role of the Fourier transform -- 5.4 The Haar wavelet -- 5.5 The role of compact support -- 5.6 Wavelets and singularities -- 5.7 Best N-term approximation -- 5.8 Frames -- 5.9 Gabor systems -- 5.10 Exercises -- Appendix A -- A.1 Definitions and notation -- A.2 Proof of Weierstrass’ theorem -- A.3 Proof of Taylor’s theorem -- A.4 Infinite series -- A.5 Proof of Theorem 3 7 2 -- Appendix B -- B.1 Power series -- B.2 Fourier series for 2?-periodic functions -- List of Symbols -- References En línea: http://dx.doi.org/10.1007/978-0-8176-4448-2 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=35195 Approximation Theory : From Taylor Polynomials to Wavelets [documento electrónico] / Christensen, Ole ; SpringerLink (Online service) ; Christensen, Khadija L . - Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2005 . - XI, 156 p. 5 illus : online resource. - (Applied and Numerical Harmonic Analysis, ISSN 2296-5009) .
ISBN : 978-0-8176-4448-2
Idioma : Inglés (eng)
Palabras clave: Mathematics Harmonic analysis Approximation theory Fourier Functional Applied mathematics Engineering Analysis Approximations and Expansions Abstract Applications of Signal, Image Speech Processing Clasificación: 51 Matemáticas Resumen: This concisely written book gives an elementary introduction to a classical area of mathematics—approximation theory—in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Key features and topics: * Description of wavelets in words rather than mathematical symbols * Elementary introduction to approximation using polynomials (Weierstrass’ and Taylor’s theorems) * Introduction to infinite series, with emphasis on approximation-theoretic aspects * Introduction to Fourier analysis * Numerous classical, illustrative examples and constructions * Discussion of the role of wavelets in digital signal processing and data compression, such as the FBI’s use of wavelets to store fingerprints * Minimal prerequisites: elementary calculus * Exercises that may be used in undergraduate and graduate courses on infinite series and Fourier series Approximation Theory: From Taylor Polynomials to Wavelets will be an excellent textbook or self-study reference for students and instructors in pure and applied mathematics, mathematical physics, and engineering. Readers will find motivation and background material pointing toward advanced literature and research topics in pure and applied harmonic analysis and related areas Nota de contenido: 1 Approximation with Polynomials -- 1.1 Approximation of a function on an interval -- 1.2 Weierstrass’ theorem -- 1.3 Taylor’s theorem -- 1.4 Exercises -- 2 Infinite Series -- 2.1 Infinite series of numbers -- 2.2 Estimating the sum of an infinite series -- 2.3 Geometric series -- 2.4 Power series -- 2.5 General infinite sums of functions -- 2.6 Uniform convergence -- 2.7 Signal transmission -- 2.8 Exercises -- 3 Fourier Analysis -- 3.1 Fourier series -- 3.2 Fourier’s theorem and approximation -- 3.3 Fourier series and signal analysis -- 3.4 Fourier series and Hilbert spaces -- 3.5 Fourier series in complex form -- 3.6 Parseval’s theorem -- 3.7 Regularity and decay of the Fourier coefficients -- 3.8 Best N-term approximation -- 3.9 The Fourier transform -- 3.10 Exercises -- 4 Wavelets and Applications -- 4.1 About wavelet systems -- 4.2 Wavelets and signal processing -- 4.3 Wavelets and fingerprints -- 4.4 Wavelet packets -- 4.5 Alternatives to wavelets: Gabor systems -- 4.6 Exercises -- 5 Wavelets and their Mathematical Properties -- 5.1 Wavelets and L2 (?) -- 5.2 Multiresolution analysis -- 5.3 The role of the Fourier transform -- 5.4 The Haar wavelet -- 5.5 The role of compact support -- 5.6 Wavelets and singularities -- 5.7 Best N-term approximation -- 5.8 Frames -- 5.9 Gabor systems -- 5.10 Exercises -- Appendix A -- A.1 Definitions and notation -- A.2 Proof of Weierstrass’ theorem -- A.3 Proof of Taylor’s theorem -- A.4 Infinite series -- A.5 Proof of Theorem 3 7 2 -- Appendix B -- B.1 Power series -- B.2 Fourier series for 2?-periodic functions -- List of Symbols -- References En línea: http://dx.doi.org/10.1007/978-0-8176-4448-2 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=35195 Ejemplares
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Título : Frames and Bases : An Introductory Course Tipo de documento: documento electrónico Autores: Christensen, Ole ; SpringerLink (Online service) Editorial: Boston : Birkhäuser Boston Fecha de publicación: 2008 Colección: Applied and Numerical Harmonic Analysis Número de páginas: XVIII, 313 p. 14 illus Il.: online resource ISBN/ISSN/DL: 978-0-8176-4678-3 Idioma : Inglés (eng) Palabras clave: Mathematics Harmonic analysis Fourier Functional Operator theory Mathematical models Analysis Modeling and Industrial Signal, Image Speech Processing Abstract Theory Clasificación: 51 Matemáticas Resumen: 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 Nota de contenido: Frames in Finite-dimensional Inner Product Spaces -- Infinite-dimensional Vector Spaces and Sequences -- Bases -- Bases and their Limitations -- Frames in Hilbert Spaces -- B-splines -- Frames of Translates -- Shift-Invariant Systems -- Gabor Frames in L(R) -- Gabor Frames in l(Z) -- Wavelet Frames in L(R) En línea: http://dx.doi.org/10.1007/978-0-8176-4678-3 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34275 Frames and Bases : An Introductory Course [documento electrónico] / Christensen, Ole ; SpringerLink (Online service) . - Boston : Birkhäuser Boston, 2008 . - XVIII, 313 p. 14 illus : online resource. - (Applied and Numerical Harmonic Analysis) .
ISBN : 978-0-8176-4678-3
Idioma : Inglés (eng)
Palabras clave: Mathematics Harmonic analysis Fourier Functional Operator theory Mathematical models Analysis Modeling and Industrial Signal, Image Speech Processing Abstract Theory Clasificación: 51 Matemáticas Resumen: 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 Nota de contenido: Frames in Finite-dimensional Inner Product Spaces -- Infinite-dimensional Vector Spaces and Sequences -- Bases -- Bases and their Limitations -- Frames in Hilbert Spaces -- B-splines -- Frames of Translates -- Shift-Invariant Systems -- Gabor Frames in L(R) -- Gabor Frames in l(Z) -- Wavelet Frames in L(R) En línea: http://dx.doi.org/10.1007/978-0-8176-4678-3 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34275 Ejemplares
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Título : Functions, Spaces, and Expansions : Mathematical Tools in Physics and Engineering Tipo de documento: documento electrónico Autores: Christensen, Ole ; SpringerLink (Online service) Editorial: Boston : Birkhäuser Boston Fecha de publicación: 2010 Colección: Applied and Numerical Harmonic Analysis Número de páginas: XIX, 266 p. 9 illus Il.: online resource ISBN/ISSN/DL: 978-0-8176-4980-7 Idioma : Inglés (eng) Palabras clave: Mathematics Fourier analysis Functional Special functions Mathematical models Physics Applied mathematics Engineering Modeling and Industrial Analysis Appl.Mathematics/Computational Methods of in Functions Clasificación: 51 Matemáticas Resumen: This graduate-level textbook is a detailed exposition of key mathematical tools in analysis aimed at students, researchers, and practitioners across science and engineering. Every topic covered has been specifically chosen because it plays a key role outside the field of pure mathematics. Although the treatment of each topic is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are fundamental to exploring the computational aspects of physics and engineering. A central theme of the book is the structure of various vector spaces—most importantly, Hilbert spaces—and expansions of elements in these spaces in terms of bases. Key topics and features include: * More than 150 exercises * Abstract and normed vector spaces * Approximation in normed vector spaces * Hilbert and Banach spaces * General bases and orthonormal bases * Linear operators on various normed spaces * The Fourier transform, including the discrete Fourier transform * Wavelets and multiresolution analysis * B-splines * Sturm–Liouville problems As a textbook that provides a deep understanding of central issues in mathematical analysis, Functions, Spaces, and Expansions is intended for graduate students, researchers, and practitioners in applied mathematics, physics, and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. Familiarity with the basic concepts of calculus and real analysis, including Riemann integrals and infinite series of real or complex numbers, is also required. Functions, Spaces, and Expansions is the main textbook for the e-course Mathematics 4: Real Analysis currently being taught at the Technical University of Denmark. Please click the "Course Materials" link on the right to access videos of the lectures, problem sheets, and solutions to selected exercises Nota de contenido: Mathematical Background -- Normed Vector Spaces -- Banach Spaces -- Hilbert Spaces -- The Lp-spaces -- The Hilbert Space L2 -- The Fourier Transform -- An Introduction to Wavelet Analysis -- A Closer Look at Multiresolution Analysis -- B-splines -- Special Functions -- Appendix A -- Appendix B En línea: http://dx.doi.org/10.1007/978-0-8176-4980-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33564 Functions, Spaces, and Expansions : Mathematical Tools in Physics and Engineering [documento electrónico] / Christensen, Ole ; SpringerLink (Online service) . - Boston : Birkhäuser Boston, 2010 . - XIX, 266 p. 9 illus : online resource. - (Applied and Numerical Harmonic Analysis) .
ISBN : 978-0-8176-4980-7
Idioma : Inglés (eng)
Palabras clave: Mathematics Fourier analysis Functional Special functions Mathematical models Physics Applied mathematics Engineering Modeling and Industrial Analysis Appl.Mathematics/Computational Methods of in Functions Clasificación: 51 Matemáticas Resumen: This graduate-level textbook is a detailed exposition of key mathematical tools in analysis aimed at students, researchers, and practitioners across science and engineering. Every topic covered has been specifically chosen because it plays a key role outside the field of pure mathematics. Although the treatment of each topic is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are fundamental to exploring the computational aspects of physics and engineering. A central theme of the book is the structure of various vector spaces—most importantly, Hilbert spaces—and expansions of elements in these spaces in terms of bases. Key topics and features include: * More than 150 exercises * Abstract and normed vector spaces * Approximation in normed vector spaces * Hilbert and Banach spaces * General bases and orthonormal bases * Linear operators on various normed spaces * The Fourier transform, including the discrete Fourier transform * Wavelets and multiresolution analysis * B-splines * Sturm–Liouville problems As a textbook that provides a deep understanding of central issues in mathematical analysis, Functions, Spaces, and Expansions is intended for graduate students, researchers, and practitioners in applied mathematics, physics, and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. Familiarity with the basic concepts of calculus and real analysis, including Riemann integrals and infinite series of real or complex numbers, is also required. Functions, Spaces, and Expansions is the main textbook for the e-course Mathematics 4: Real Analysis currently being taught at the Technical University of Denmark. Please click the "Course Materials" link on the right to access videos of the lectures, problem sheets, and solutions to selected exercises Nota de contenido: Mathematical Background -- Normed Vector Spaces -- Banach Spaces -- Hilbert Spaces -- The Lp-spaces -- The Hilbert Space L2 -- The Fourier Transform -- An Introduction to Wavelet Analysis -- A Closer Look at Multiresolution Analysis -- B-splines -- Special Functions -- Appendix A -- Appendix B En línea: http://dx.doi.org/10.1007/978-0-8176-4980-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33564 Ejemplares
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