Document Zbl 1481.42007

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Funct. Anal.</a> <a href="/?q=in%3A472063" title="Articles in this Issue">282, No. 1, Article ID 109272, 62 p. (2022)</a>. </div> <div class="abstract">It is known that several classical results that hold for one dimensinal Fourier series fail for high dimensional Fourier series. For example, the Fourier series of the indicator function of a $d$-dimensional ball with $d\geq 5$, the spherical partial sum diverges at all rational points, while it converges almost everywhere. This was proved by using results on the lattice point problems. The main results in this paper are to study some singular phenomena and convergence properties of multiple Fourier series. Theorem 1.1 and Corollary 1.2 deal with the behavior of $S_\lambda(u_{\beta,a})$ at $x = 0$ including the Pinsky phenomenon, Theorem 1.3 deals with the pointwise behaviors, Theorem 1.4 deals with the Gibbs-Wilbraham phenomenon near $G_a$, and, Theorem 1.5 deals with the almost everywhere convergence.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A14646">Chun-Yen Shen (Taipei)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7424007">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A42B05" title="MSC2020">42B05</a> </td> <td class="space"> Fourier series and coefficients in several variables </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AGibbs-Wilbraham+phenomenon">Gibbs-Wilbraham phenomenon</a>; <a href="/?q=ut%3APinsky+phenomenon">Pinsky phenomenon</a>; <a href="/?q=ut%3Aspherical+partial+sum">spherical partial sum</a>; <a href="/?q=ut%3Aradial+function">radial function</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 1481.42007" data-ciurl="/ci/07424007" data-biburl="/bibtex/07424007.bib" data-amsurl="/amsrefs/07424007.bib" data-xmlurl="/xml/07424007.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07424007.pdf" title="Zbl 1481.42007 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.jfa.2021.109272" aria-label="DOI for “Multiple Fourier series and lattice point problems”" title="10.1016/j.jfa.2021.109272">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2011.06178"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">Ashurov, R., Generalized localization for spherical partial sums of multiple Fourier series, J. 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