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Boston, MA: Birkh盲user. xv, 359 p. (2001). </div> <div class="abstract">This book is written by one of the leading experts in Gabor analysis and deserves considerably interest. It gives a unified approach to most of the modern theory for time-frequency analysis from a mathematician’s point of view, with new proofs of many recent results. The basic idea in time-frequency analysis is to represent a signal as a superposition of shifted and modulated versions of a single function (the Gabor window). The goal is to obtain representations that combine information about the signal and its Fourier transform (within the limitations arising from the uncertainty principle).<br class="zbmathjax-paragraph">The starting point is basic Fourier analysis, which is the subject of Chapter 1. Chapter 2 deals with different versions of the uncertainty principle. Chapter 3 contains the basic results about the short-time Fourier transform (also called the window Fourier transform), in particular, the orthogonality relations and the inversion formula. Chapter 4 is devoted to quadratic time-frequency representations (Wigner distributions and the ambiguity function), while Gabor frames are treated in Chapter 5-7. Besides the fundamental results about frames, a new approach to many important results by Daubechies, Janssen and Ron/Shen is presented. Different representations of the Gabor frame operator (due to Janssen and Walnut) are discussed and sufficient conditions for a Gabor family to be a frame are given. In Chapter 8 the topic is the Zak transform and its role in connection with Gabor frames. A new approach to Wilson bases can be found here. Several versions of Balian-Low’s theorem are proved. The last chapters present more advanced topics, like a discussion of the non-commutative Heisenberg group and representation theory. Time-frequency analysis is extended to various Banach spaces, including modulation spaces. The role of Feichtinger’s algebra \(M^1\) is highlighted. A short Chapter describes the similarities and differences between Gabor analysis and wavelet analysis. In both cases, the relevant transforms are representation coefficients associated to an integrable unitary group representation and satisfy the orthogonality relations. On the other hand, most of the results about the structure of Gabor frames have (at least at the moment) no analogue for wavelet frames. The last Chapter deals with pseudodifferential operators and representations of them in terms of time-frequency shifts.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A9520">Ole Christensen (Lyngby)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1554752">Cited in <strong>12</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A1554752">Cited in <strong>1241</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C40" title="MSC2020">42C40</a> </td> <td class="space"> Nontrigonometric harmonic analysis involving wavelets and other special systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A12" title="MSC2020">94A12</a> </td> <td class="space"> Signal theory (characterization, reconstruction, filtering, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42-02" title="MSC2020">42-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94-02" title="MSC2020">94-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to information and communication theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A43-02" title="MSC2020">43-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AGabor+analysis">Gabor analysis</a>; <a href="/?q=ut%3Atime-frequency+analysis">time-frequency analysis</a>; <a href="/?q=ut%3Auncertainty+principle">uncertainty principle</a>; <a href="/?q=ut%3Ashort-time+Fourier+transform">short-time Fourier transform</a>; <a href="/?q=ut%3Awindow+Fourier+transform">window Fourier transform</a>; <a href="/?q=ut%3AGabor+frames">Gabor frames</a>; <a href="/?q=ut%3AZak+transform">Zak transform</a>; <a href="/?q=ut%3AWilson+bases">Wilson bases</a>; <a href="/?q=ut%3Arepresentation+theory">representation theory</a>; <a href="/?q=ut%3Awavelet+analysis">wavelet analysis</a>; <a href="/?q=ut%3Apseudodifferential+operators">pseudodifferential operators</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0966.42020" data-ciurl="/ci/01554752" data-biburl="/bibtex/01554752.bib" data-amsurl="/amsrefs/01554752.bib" data-xmlurl="/xml/01554752.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01554752.pdf" title="Zbl 0966.42020 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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