Document JFM 65.0278.03

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Math. Soc., Colloquium Publ., Vol. 23. New York. American Mathematical Society. vii, 401 p. (1939).</a> </div> <div class="abstract">Das inhaltsreiche Werk bringt erstmalig eine ausf眉hrliche zusammenfassende 脺bersicht 眉ber das gesamte Gebiet der Orthogonalpolynome und ist als Lehrbuch wie als Nachschlagewerk gleicherma脽en geeignet. Ausf眉hrliche Herleitung wechselt mit kurzer Angabe der Ergebnisse. Mancherlei ist hier erstmalig ver枚ffentlicht, zahlreiche Untersuchungen und Beweise sind neu oder wenigstens neuartig dargestellt. Dem speziellen Interesse des Verf. und seinen eigenen Untersuchungen entsprechend liegt das Hauptgewicht in der Stoffauswahl weniger auf den formalen Eigenschaften der Orthogonalpolynome als auf den analytischen; Identit盲ten, Integraldarstellungen usw. sind also nicht so ausf眉hrlich behandelt wie asymptotische Entwicklungen, Summierung von Polynomreihen u. a. Rein algebraische Fragestellungen (Irreduzibilit盲t usw.) sind nicht aufgenommen. 脺ber Polynome, die auf beliebigen Kurven orthogonal sind, ist alles Bisherige zusammengestellt und erweitert.<br class="zbmathjax-paragraph">Die einzelnen Kapitel k枚nnen auch f眉r sich gelesen werden; sie enthalten den Stoff in folgender Anordnung: Kap. I-III: Hilfss盲tze und -begriffe, Definitionen und Beispiele, die sogenannten “klassischen” Orthogonalpolynome, allgemeine und formale Eigenschaften. Ausgangspunkt dieser Entwicklungen ist dabei die Orthogonalit盲t. Kap. IV und V: Formale Eigenschaften der klassischen Polynome, Darstellungen, Zusammenh盲nge mit andern Funktionen, die “zweiten L枚sungen” der Differentialgleichungen. VI: Absch盲tzungen f眉r die Nullstellen der Orthogonalpolynome, besonders f眉r die klassischen. Die Nullstellen der Jacobischen Polynome im nichtorthogonalen Fall. Kap. VII und VIII: Ungleichungen f眉r die Orthogonalpolynome, asymptotische Entwicklungen f眉r die klassischen Polynome mit besonders ausf眉hrlichen Darstellungen der verschiedenen Methoden. Kap. IX und X: Hilfss盲tze, Aussagen 眉ber die nach den klassischen Polynomen fortschreitenden Reihen, ihre Konvergenz und Ces脿rosummabilit盲t, die Verh盲ltnisse an den Enden des Intervalls, alles mit ausf眉hrlichen Beweisen. Die Entwicklung analytischer Funktionen. Kap. XI: Polynome, die auf dem Einheitskreis orthogonal sind; allgemeine Eigenschaften und Beziehungen zum reellen Fall. Kap. XII und XIII: Asymptotische Aussagen und Reihenentwicklungen im allgemeinen Fall beliebiger Orthogonalpolynome. Kap. XIV: Interpolation mit Nullstellen von Orthogonalpolynomen als Abszissen; Lagrangesche Interpolation, Fej茅rsche “Treppenpolynome”, Konvergenzfragen. Kap. XV: Mechanische Quadraturen auf klassischen Abszissen, Cotessche Zahlen. Kap. XVI: Polynome, die auf beliebigen Kurven orthogonal sind (allgemeines, asymptotisches Verhalten, Reihen).<br class="zbmathjax-paragraph">Sechzig Aufgaben und 脺bungen im Stile des “P贸lya-Szeg枚” (<span class="zbmathjax-textit">G. P贸lya, G. Szeg枚</span>, Aufgaben und Lehrs盲tze aus der Analysis (1925); F. d. M. 51, 173) dienen zur Vertiefung und bringen mancherlei Nebenbeweise und -ergebnisse. – Das Literaturverzeichnis, in dem wohl nur einige unwichtige und entlegene Arbeiten fehlen, umfa脽t 325 Nummern.<div class="reviewer"> Reviewer: <a href="/authors/?q=Hahn+W%2A">Hahn, W., Dr. (Berlin)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A2510415">Cited in <strong>5</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A2510415">Cited in <strong>1672</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A33-02" title="MSC2020">33-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to special functions </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></table> </div><div class="keywords"> <h3>JFM Section:</h3><a href="/?q=ut%3A%22Erster+Halbband.+D.+Analysis.+7.+Besondere+Funktionen+und+Funktionensysteme.+b%29+Besondere+Orthogonalpolynome."">Erster Halbband. D. Analysis. 7. Besondere Funktionen und Funktionensysteme. b) Besondere Orthogonalpolynome.</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="JFM" data-itemname="JFM 65.0278.03" data-ciurl="" data-biburl="/bibtex/02510415.bib" data-amsurl="/amsrefs/02510415.bib" data-xmlurl="/xml/02510415.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02510415.pdf" title="JFM 65.0278.03 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 = ""; </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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