Document Zbl 1365.65294

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II: Algebraic, stochastic and analysis structures for networks, data classification and optimization. Cham: Springer (ISBN 978-3-319-42104-9/hbk; 978-3-319-42105-6/ebook). Springer Proceedings in Mathematics & Statistics 179, 337-353 (2016). </div> <div class="abstract">The Ortigueira-Caputo fractional operator, which provides the fractional derivative of a complex function, is rewritten in the distribution sense. The fractional derivative of the Gabor-Morlet wavelet is computed. An open problem, concerning the possibility to generalize the reconstruction formula for Shannon wavelets through the distribution theory, is proposed.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1365.74007">Zbl 1365.74007</a>].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A13617">Yuri A. Farkov (Moscow)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6703854">Cited in <strong>99</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65T60" title="MSC2020">65T60</a> </td> <td class="space"> Numerical methods for wavelets </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Awavelet+basis">wavelet basis</a>; <a href="/?q=ut%3Apositive+definite+distribution">positive definite distribution</a>; <a href="/?q=ut%3Acomplex+fractional+derivative">complex fractional derivative</a>; <a href="/?q=ut%3AGabor-Morlet+wavelet">Gabor-Morlet wavelet</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 1365.65294" data-ciurl="/ci/06703854" data-biburl="/bibtex/06703854.bib" data-amsurl="/amsrefs/06703854.bib" data-xmlurl="/xml/06703854.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06703854.pdf" title="Zbl 1365.65294 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/10.1007/978-3-319-42105-6_16" aria-label="DOI for “Fractional-wavelet analysis of positive definite distributions and wavelets on ${\mathcal D'(\mathbb {C})}$”" title="10.1007/978-3-319-42105-6_16">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Martino Fine Books, New York (2014) · <a href="/0171.38503" class="nowrap">Zbl 0171.38503</a></td> </tr><tr> <td>[2]</td> <td class="space">Cattani, C.: Shannon wavelets theory. Math. Probl. Eng. 2008, 1–24 (2008) · <a href="/1162.42314" class="nowrap">Zbl 1162.42314</a></td> </tr><tr> <td>[3]</td> <td class="space">Corinthios, M.J.: Generalisation of the Dirac-delta impulse extending Laplace and z transform domains. IEE Proc. Vis. Image Sig. Process. 150, 69–81 (2003) · <a href="https://doi.org/10.1049/ip-vis:20030236" class="nowrap">doi:10.1049/ip-vis:20030236</a></td> </tr><tr> <td>[4]</td> <td class="space">Corinthios, M.J.: Complex-variable distribution theory for Laplace and z transforms. IEE Proc. Vis. Image Sig. Process. 152, 97–106 (2005) · <a href="https://doi.org/10.1049/ip-vis:20050999" class="nowrap">doi:10.1049/ip-vis:20050999</a></td> </tr><tr> <td>[5]</td> <td class="space">Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992) · <a href="/0776.42018" class="nowrap">Zbl 0776.42018</a> · <a href="https://doi.org/10.1137/1.9781611970104" class="nowrap">doi:10.1137/1.9781611970104</a></td> </tr><tr> <td>[6]</td> <td class="space">Donoghue Jr., W.F.: Distributions and Fourier Transform. Academic Press, New York (1969)</td> </tr><tr> <td>[7]</td> <td class="space">Gabardo, J.-P.: Extension of Positive-Definite Distributions and Maximum Entropy. Memoirs of the American Mathematical Society, vol. 102. American Mathematical Society, Providence (1993)</td> </tr><tr> <td>[8]</td> <td class="space">Gel’fand, I.M., Shilov, G.E.: Generalized Functions. Academic Press, New York (1964)</td> </tr><tr> <td>[9]</td> <td class="space">Guariglia, E.: Fractional derivative of the Riemann zeta function. In: Cattani C., Srivastava H., Yang X.J. (eds.) Fractional Dynamics, De Gruyter Open, Chap. 21 (2015) · <a href="https://doi.org/10.1515/9783110472097-022" class="nowrap">doi:10.1515/9783110472097-022</a></td> </tr><tr> <td>[10]</td> <td class="space">Hernández, E., Weiss, G.: A First Course of Wavelets. CRC Press, Boca Raton (1996) · <a href="/0885.42018" class="nowrap">Zbl 0885.42018</a> · <a href="https://doi.org/10.1201/9781420049985" class="nowrap">doi:10.1201/9781420049985</a></td> </tr><tr> <td>[11]</td> <td class="space">Hölschneider, M.: Wavelets: An Analysis Tool. Clarendon Press, New York (1999)</td> </tr><tr> <td>[12]</td> <td class="space">Lemarié, P.G., Meyer, Y.: Ondeletles et bases hilbertiennes. Revista Matemática Iberoamericana 2, 1–18 (1986) · <a href="/0657.42028" class="nowrap">Zbl 0657.42028</a> · <a href="https://doi.org/10.4171/RMI/22" class="nowrap">doi:10.4171/RMI/22</a></td> </tr><tr> <td>[13]</td> <td class="space">Li, C., Dao, X., Guo, P.: Fractional derivatives in complex planes. Nonlinear Anal. Theory Methods Appl. 71, 1857–1869 (2009) · <a href="/1173.26305" class="nowrap">Zbl 1173.26305</a> · <a href="https://doi.org/10.1016/j.na.2009.01.021" class="nowrap">doi:10.1016/j.na.2009.01.021</a></td> </tr><tr> <td>[14]</td> <td class="space">Neeb, K.-H., Ólafsson, G.: Reflection positivity and conformal symmetry. J. Funct. Anal. Elsevier 266, 2174–2224 (2014) · <a href="/1290.22006" class="nowrap">Zbl 1290.22006</a> · <a href="https://doi.org/10.1016/j.jfa.2013.10.030" class="nowrap">doi:10.1016/j.jfa.2013.10.030</a></td> </tr><tr> <td>[15]</td> <td class="space">Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers, pp. 35–37. Springer, London (2011) · <a href="/1251.26005" class="nowrap">Zbl 1251.26005</a> · <a href="https://doi.org/10.1007/978-94-007-0747-4" class="nowrap">doi:10.1007/978-94-007-0747-4</a></td> </tr><tr> <td>[16]</td> <td class="space">Saneva, K., Vindas, J.: Wavelet expansions and asymptotic behaviour of distributions. J. Math. Anal. Appl. 370, 543–554 (2010) · <a href="/1196.42031" class="nowrap">Zbl 1196.42031</a> · <a href="https://doi.org/10.1016/j.jmaa.2010.04.041" class="nowrap">doi:10.1016/j.jmaa.2010.04.041</a></td> </tr><tr> <td>[17]</td> <td class="space">Schwartz, L.: Théorie des Distributions, vol. 1-2. Hermann, Paris (1951)</td> </tr><tr> <td>[18]</td> <td class="space">Teolis, A.: Computational Signal Processing with Wavelets. Birkhäuser, Boston (1998) · <a href="/0928.94002" class="nowrap">Zbl 0928.94002</a></td> </tr><tr> <td>[19]</td> <td class="space">Triebel, H.: Function Spaces and Wavelets on Domains. European Mathematical Society, Berlin (2008) · <a href="/1158.46002" class="nowrap">Zbl 1158.46002</a> · <a href="https://doi.org/10.4171/019" class="nowrap">doi:10.4171/019</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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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