A Basis Theory Primer

This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and their use in both applied and classical harmonic analysis. The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory.
Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses. The self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications.

This textbook is a mathematical introduction to the theory of bases in Banach spaces, also weaving in the theory of bases in Hilbert spaces and generalizations of bases such as frames. A new impetus for the study of bases in Banach spaces has come from the rise of wavelet theory, and especially the theory of frames, which allow basis-like but possibly redundant expansions.This is a text to learn from, with detailed explanations of abstract material that are usually explained with the remark "clearly ..." Covering abstract material with a high degree of relevance to a wide variety of modern topics, the book is primarily aimed at mathematicians, but will also encourage engineers to venture into new directions. Exercises at the end of each chapter make the work suitable as a textbook for a variety of graduate-level courses on bases and frames.

Unique book in the literature A clear and accessible text with detailed explanations of abstract material Suitable for classroom use or independent study Covers abstract material with a high degree of relevance to a wide range of modern topics Written for a broad audience of graduate students, pure and applied mathematicians as well as engineers Includes extensive exercises at the end of each section Separate solutions manual available for instructors upon request Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material

Inhalt

ANHA Series Preface.- Preface.- General Notation.- Part I. A Primer on Functional Analysis .- Banach Spaces and Operator Theory.- Functional Analysis.- Part II. Bases and Frames .- Unconditional Convergence of Series in Banach and Hilbert Spaces.- Bases in Banach Spaces.- Biorthogonality, Minimality, and More About Bases.- Unconditional Bases in Banach Spaces.- Bessel Sequences and Bases in Hilbert Spaces.- Frames in Hilbert Spaces.- Part III. Bases and Frames in Applied Harmonic Analysis .- The Fourier Transform on the Real Line.- Sampling, Weighted Exponentials, and Translations.- Gabor Bases and Frames.- Wavelet Bases and Frames.- Part IV. Fourier Series .- Fourier Series.- Basic Properties of Fourier Series.- Part V. Appendices .- Lebesgue Measure and Integration.- Compact and HilbertSchmidt Operators.- Hints for Exercises.- Index of Symbols.- References.- Index.