Document Zbl 1486.47086

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French summary)</i> <a class="label nowrap" href="/1486.47086">Zbl 1486.47086</a> </h2> <div class="source"> <a href="/serials/1597" title="Journal Profile">J. Math. Pures Appl. (9)</a> <a href="/?q=in%3A414479" title="Articles in this Issue">118, 288-316 (2018)</a>. </div> <div class="abstract">Summary: Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We link the Hausdorff-Young inequality for the Fourier-Wigner transform with Lieb’s inequality for ambiguity functions. Noncommutative Tauberian theorems due to <span class="zbmathjax-textit">R. Werner</span> [J. Math. Phys. 25, 1404–1411 (1984; <a href="/0557.43003">Zbl 0557.43003</a>)] allow us to extend results of <span class="zbmathjax-textit">D. Bayer</span> and <span class="zbmathjax-textit">K. Gröchenig</span> [Integral Equations Oper. Theory 82, No. 1, 95–117 (2015; <a href="/1337.47029">Zbl 1337.47029</a>)] on localization operators. Furthermore, we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6938398">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6938398">Cited in <strong>31</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47G30" title="MSC2020">47G30</a> </td> <td class="space"> Pseudodifferential operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35S05" title="MSC2020">35S05</a> </td> <td class="space"> Pseudodifferential operators as generalizations of partial differential operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E35" title="MSC2020">46E35</a> </td> <td class="space"> Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47B10" title="MSC2020">47B10</a> </td> <td class="space"> Linear operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alocalization+operators">localization operators</a>; <a href="/?q=ut%3ABerezin+transform">Berezin transform</a>; <a href="/?q=ut%3ABanach+modules">Banach modules</a>; <a href="/?q=ut%3AArveson+spectrum">Arveson spectrum</a>; <a href="/?q=ut%3ATauberian+theorems">Tauberian theorems</a>; <a href="/?q=ut%3Amodulation+spaces">modulation spaces</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0557.43003">Zbl 0557.43003</a>; <a href="/1337.47029">Zbl 1337.47029</a></div>  <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;">  </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">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1486.47086" data-ciurl="/ci/06938398" data-biburl="/bibtex/06938398.bib" data-amsurl="/amsrefs/06938398.bib" data-xmlurl="/xml/06938398.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06938398.pdf" title="Zbl 1486.47086 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.matpur.2017.12.004" aria-label="DOI for “Convolutions for localization operators”" title="10.1016/j.matpur.2017.12.004">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1705.03253"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Arveson, W. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </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 = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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