Mathematical Tools in Physics and Engineering
Focusing on essential mathematical tools in analysis, this text emphasizes their significance beyond pure mathematics. It highlights topics that are particularly relevant for applications in computational physics and engineering, making it a valuable resource for those looking to bridge theoretical concepts with practical use in various scientific fields.
This revised and expanded monograph presents a comprehensive theory for frames and Riesz bases in Hilbert spaces, along with practical applications in Gabor analysis, wavelet analysis, and generalized shift-invariant systems. The second edition emphasizes explicit constructions with appealing properties and includes new sections on LCA groups, generalized shift-invariant systems, and duality theory for Gabor and wavelet frames, reflecting a decade of advancements in frame theory. Key features include an elementary introduction to frame theory in finite-dimensional spaces, accessible basic results for both pure and applied mathematicians, and extensive exercises suitable for graduate courses. Full proofs are provided in introductory chapters, requiring only a basic understanding of functional analysis. The book discusses explicit constructions of frames and dual pairs, with applications to time-frequency analysis and connections to sampling theory. Selected research topics and recommendations for further reading are included, along with open problems to encourage continued research. This work will appeal to graduate students and researchers in pure and applied mathematics, mathematical physics, and engineering, as well as professionals in digital signal processing seeking to understand the underlying theory of modern tools.
In recent years, frames have gained popularity across various applications, leading to numerous constructions based on direct methods rather than classical sufficient conditions. This has created a demand for an updated resource that shifts from traditional approaches to a more constructive perspective. This textbook addresses this need by building on the author's previous work, presenting frame theory as a dialogue between mathematicians and engineers. It includes new sections on practical applications, helping mathematically inclined readers understand real-world uses of frames while providing engineers with the mathematical foundations relevant to their fields. Key features include accessible presentations for graduate students, mathematicians, and engineers, along with an introductory chapter on finite-dimensional vector spaces that simplifies the concept of frames. Extensive exercises cater to both theoretical graduate courses and application-focused courses in Gabor analysis or wavelets. The book offers a detailed examination of frames, including full proofs and the relationship between frames and Riesz bases, while discussing various construction methods. The content is divided into two parts: the first covers abstract theory, and the second focuses on explicit constructions and their applications in time-frequency analysis, Gabor analysis, and wavelets. This resource serves as an excellent textbook for graduate studen
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.