Frames and bases

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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