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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.
Inhaltsverzeichnis
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 Hilbert Schmidt Operators. - Hints for Exercises. - Index of Symbols. - References. - Index.
From the reviews:
The present book gives a wide perspective, preparing the functional analytic ground and also discussing in great detail the relevant features of bases in Banach spaces, unconditional bases, frames, and their role in the context of Applied Harmonic Analysis. the book is ideally suited for self-study, but also as a text book from which different courses can be compiled. The presentation is very reader-friendly and provides all necessary details. (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 166 (3-4), June, 2012)
This book is a very comprehensive work dedicated to introducing graduate students or researchers in pure and applied mathematics as well as engineering to the foundations of basis expansions and to essential techniques for applications. The exercises contained in the book make it a good fit for graduate courses on selected topics in functional analysis and applications. (Bernhard Bodmann, Zentralblatt MATH, Vol. 1227, 2012)
The amount of mathematics treated in the book is impressive. a handbook for a certain group of mathematicians to learn about the main tools of the theory of bases and frames for Banach and Hilbert spaces. Personally I like this book. It is one of those very few mathematical books that I can read without additional difficulties arising from my limited capacity to remember facts and definitions. (Kazaros Kazarian, Mathematical Reviews, Issue 2012 b)
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